Advanced

Tutor model

i62 💰:9.4

i52 💰:7.9

🏆 i67 💰:62.5

i70 💰:125.0

i68 💰:125.0

🏆 i66 💰:75.0

i65 💰:125.0

i63 💰:62.5

i61 💰:75.0

i60 💰:9.4

i60 💰:37.5

i58 💰:62.5

i57 💰:22.5

i57 💰:37.5

i56 💰:45.0

⚡ i55 💰:11.25

i55 💰:37.5

i54 💰:62.5

i52 💰:35.0

i51 💰:37.5

i49 💰:15.0

i49 💰:62.5

i48 💰:11.3

i47 💰:7.5

i44 💰:25.0

i43 💰:37.5

i42 💰:187.5

i40 💰:5.0

i39 💰:100.0

i38 💰:37.5

i37 💰:11.0

i36 💰:187.5

i34 💰:12.5

i33 💰:37.5

i30 💰:1.0

i29 💰:1.5

i28 💰:150.0

i23 💰:20.0

i20 💰:4.0

i18 💰:1.0

i16 💰:25.0

i13 💰:1.5

Keep session private

What would you like to learn?

Override for the tutor's lesson content generation calls. Auto uses the system default (Claude Sonnet 5 for lesson text, Opus 5 for diagrams).

Lesson generation model

System default

i58 💰:18.8

i31 💰:11.0

i25 💰:31.3

Tutoring abilities

If you connect three light bulbs in a single line across a 9-volt battery, every bulb glows noticeably dimmer than if you connected just one. Why does stringing components back-to-back force the total opposition to current to stack up?

In a **series circuit**, electrical charge has only one continuous pathway to travel, meaning every coulomb of charge that leaves the battery must pass through every single resistor in sequence.

According to **Kirchhoff's Second Law** (the **Loop Rule**), conservation of energy dictates that around any closed loop, the sum of electromotive forces equals the sum of potential differences: $\sum \mathcal{E} = \sum V$.

How do we turn this physical law of energy conservation into a mathematical formula for total resistance?

**Kirchhoff's First Law** (conservation of charge) dictates that in an unbranched series circuit, charge cannot accumulate or vanish, keeping current constant everywhere: $I_{\text{total}} = I_1 = I_2 = \dots = I_n = I$.

<!DOCTYPE html>
<html lang="en">
<head>
  <meta charset="UTF-8">
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  <meta name="aspect-ratio" content="4:3">
  <link rel="stylesheet" href="{{VENDOR_CDN}}/katex.min.css" />
  <script src="{{VENDOR_CDN}}/katex.min.js"></script>
  <script src="{{VENDOR_CDN}}/classace-diagram.js"></script>
  <style>
    * { box-sizing: border-box; margin: 0; padding: 0; }
    body {
      font-family: system-ui, -apple-system, sans-serif;
      color: #1e2945;
      background: #ffffff;
      padding: 12px;
      display: flex;
      flex-direction: column;
      gap: 10px;
      user-select: none;
    }
    
    .nav-bar {
      display: flex;
      justify-content: space-between;
      align-items: center;
      gap: 8px;
    }
    .btn-group {
      display: flex;
      gap: 4px;
      background: #f1f5f9;
      padding: 3px;
      border-radius: 8px;
    }
    .step-tab {
      background: transparent;
      border: none;
      padding: 4px 10px;
      font-size: 13px;
      font-weight: 600;
      color: #64748b;
      border-radius: 6px;
      cursor: pointer;
      transition: all 0.2s ease;
    }
    .step-tab.active {
      background: #ffffff;
      color: #0284c7;
      box-shadow: 0 1px 3px rgba(0,0,0,0.1);
    }
    .play-btn {
      background: #f1f5f9;
      border: none;
      padding: 4px 10px;
      font-size: 13px;
      font-weight: 600;
      border-radius: 6px;
      cursor: pointer;
      color: #1e2945;
      display: flex;
      align-items: center;
      gap: 4px;
    }

    /* Circuit SVG area */
    .circuit-container {
      width: 100%;
      height: 140px;
      background: #f8fafc;
      border: 1px solid #e2e8f0;
      border-radius: 10px;
      position: relative;
      overflow: hidden;
    }
    svg {
      width: 100%;
      height: 100%;
      display: block;
    }

    /* Derivation Stage Display */
    .derivation-card {
      background: #f8fafc;
      border: 1px solid #e2e8f0;
      border-radius: 10px;
      padding: 12px 14px;
      min-height: 120px;
      display: flex;
      flex-direction: column;
      justify-content: center;
      align-items: center;
      gap: 8px;
      text-align: center;
    }
    .math-display {
      font-size: 19px;
      line-height: 1.4;
      color: #0f172a;
    }
    .badge {
      display: inline-block;
      padding: 3px 8px;
      background: #e0f2fe;
      color: #0369a1;
      border-radius: 6px;
      font-size: 12px;
      font-weight: 600;
    }
    
    /* Comparison banner */
    .summary-row {
      display: grid;
      grid-template-columns: 1fr 1fr;
      gap: 8px;
    }
    .info-tile {
      background: #ffffff;
      border: 1px solid #e2e8f0;
      border-radius: 8px;
      padding: 8px;
      text-align: center;
      font-size: 12px;
      color: #475569;
    }
    .info-tile strong {
      display: block;
      font-size: 14px;
      color: #0284c7;
      margin-top: 2px;
    }

    /* Current dots animation */
    .dot-flow {
      stroke-dasharray: 6 12;
      animation: flow 2s linear infinite;
    }
    @keyframes flow {
      to { stroke-dashoffset: -36; }
    }
  </style>
</head>
<body>

  
  <div class="nav-bar">
    <div class="btn-group">
      <button class="step-tab active" data-step="0">1. Current $I$</button>
      <button class="step-tab" data-step="1">2. Voltages</button>
      <button class="step-tab" data-step="2">3. Ohm's Law</button>
      <button class="step-tab" data-step="3">4. Result</button>
    </div>
    <button class="play-btn" id="playBtn"><span>⏸</span> Auto</button>
  </div>

  
  <div class="circuit-container">
    <svg viewBox="0 0 400 130" preserveAspectRatio="xMidYMid meet">
      
      <path d="M 50,65 L 50,25 L 350,25 L 350,105 L 50,105 L 50,65" fill="none" stroke="#cbd5e1" stroke-width="4" stroke-linejoin="round" />
      
      
      <path id="currentFlow" d="M 50,65 L 50,25 L 350,25 L 350,105 L 50,105 Z" fill="none" stroke="#38bdf8" stroke-width="2" class="dot-flow" />

      
      <line x1="50" y1="52" x2="50" y2="78" stroke="#cbd5e1" stroke-width="4" />
      <line x1="38" y1="56" x2="62" y2="56" stroke="#0284c7" stroke-width="3" stroke-linecap="round" />
      <line x1="43" y1="72" x2="57" y2="72" stroke="#64748b" stroke-width="4" stroke-linecap="round" />
      <text x="24" y="68" font-size="12" font-weight="700" fill="#0284c7">V</text>

      
      <rect id="boxR1" x="90" y="16" width="55" height="18" rx="3" fill="#ffffff" stroke="#0284c7" stroke-width="2" />
      <text x="117" y="29" font-size="11" font-weight="600" text-anchor="middle" fill="#0f172a">R₁</text>

      
      <rect id="boxR2" x="175" y="16" width="55" height="18" rx="3" fill="#ffffff" stroke="#0284c7" stroke-width="2" />
      <text x="202" y="29" font-size="11" font-weight="600" text-anchor="middle" fill="#0f172a">R₂</text>

      
      <rect id="boxR3" x="260" y="16" width="55" height="18" rx="3" fill="#ffffff" stroke="#0284c7" stroke-width="2" />
      <text x="287" y="29" font-size="11" font-weight="600" text-anchor="middle" fill="#0f172a">R₃</text>

      
      <g id="voltDrops" opacity="0.4" style="transition: opacity 0.3s ease;">
        <path d="M 90,40 L 145,40" stroke="#0284c7" stroke-width="1.5" stroke-dasharray="2 2" />
        <text x="117" y="52" font-size="10" font-weight="600" text-anchor="middle" fill="#0284c7">V₁</text>

        <path d="M 175,40 L 230,40" stroke="#0284c7" stroke-width="1.5" stroke-dasharray="2 2" />
        <text x="202" y="52" font-size="10" font-weight="600" text-anchor="middle" fill="#0284c7">V₂</text>

        <path d="M 260,40 L 315,40" stroke="#0284c7" stroke-width="1.5" stroke-dasharray="2 2" />
        <text x="287" y="52" font-size="10" font-weight="600" text-anchor="middle" fill="#0284c7">V₃</text>
      </g>

      
      <g id="currentLabel">
        <circle cx="200" cy="105" r="9" fill="#0284c7" />
        <text x="200" y="109" font-size="11" font-weight="700" fill="#ffffff" text-anchor="middle">I</text>
      </g>
    </svg>
  </div>

  
  <div class="derivation-card">
    <div id="stepBadge" class="badge">Conservation of Charge</div>
    <div id="mathTarget" class="math-display"></div>
  </div>

  
  <div class="summary-row">
    <div class="info-tile">
      Series Current
      <strong id="sumI">I = constant</strong>
    </div>
    <div class="info-tile">
      Equivalent Resistance
      <strong id="sumR">R_total = R₁ + R₂ + R₃</strong>
    </div>
  </div>

  <script>
    const steps = [
      {
        badge: "Kirchhoff's 1st Law (Constant Current)",
        math: "I_{\\text{total}} = I_1 = I_2 = I_3 = I",
        voltOpacity: "0.3",
        highlightI: true
      },
      {
        badge: "Kirchhoff's 2nd Law (Voltage Conservation)",
        math: "V = V_1 + V_2 + V_3",
        voltOpacity: "1.0",
        highlightI: false
      },
      {
        badge: "Apply Ohm's Law (V = IR)",
        math: "I R_{\\text{total}} = I R_1 + I R_2 + I R_3",
        voltOpacity: "1.0",
        highlightI: false
      },
      {
        badge: "Divide by Common Factor I",
        math: "R_{\\text{total}} = R_1 + R_2 + R_3",
        voltOpacity: "1.0",
        highlightI: false
      }
    ];

    let currentStep = 0;
    let autoInterval = null;
    const tabs = document.querySelectorAll('.step-tab');
    const playBtn = document.getElementById('playBtn');

    function renderStep(idx) {
      currentStep = idx;
      tabs.forEach((t, i) => t.classList.toggle('active', i === idx));
      
      const s = steps[idx];
      document.getElementById('stepBadge').textContent = s.badge;
      
      if (window.katex) {
        katex.render(s.math, document.getElementById('mathTarget'), { displayMode: true, throwOnError: false });
        katex.render("I_{\\text{total}} = I", document.getElementById('sumI'), { throwOnError: false });
        katex.render("R_{\\text{total}} = \\sum R_i", document.getElementById('sumR'), { throwOnError: false });
      } else {
        document.getElementById('mathTarget').textContent = s.math;
      }

      document.getElementById('voltDrops').style.opacity = s.voltOpacity;
    }

    function nextStep() {
      renderStep((currentStep + 1) % steps.length);
    }

    function startAuto() {
      if (autoInterval) clearInterval(autoInterval);
      autoInterval = setInterval(nextStep, 3500);
      playBtn.innerHTML = '<span>⏸</span> Pause';
    }

    function stopAuto() {
      if (autoInterval) {
        clearInterval(autoInterval);
        autoInterval = null;
      }
      playBtn.innerHTML = '<span>▶</span> Play';
    }

    tabs.forEach(tab => {
      tab.addEventListener('click', () => {
        stopAuto();
        renderStep(parseInt(tab.dataset.step, 10));
      });
    });

    playBtn.addEventListener('click', () => {
      if (autoInterval) {
        stopAuto();
      } else {
        startAuto();
      }
    });

    // Initialize
    window.addEventListener('load', () => {
      renderStep(0);
      startAuto();
    });
  </script>
</body>
</html>

Now let's follow the electric potential around the entire loop.

Because energy is conserved, the total **potential difference** $V_{\text{total}}$ provided by the power supply equals the sum of the individual potential drops across each resistor.

$$V_{\text{total}} = V_1 + V_2 + \dots + V_n$$

**Applying Ohm's law** ($V = IR$) to each individual resistor gives $V_1 = I R_1$, $V_2 = I R_2$, $\dots$, $V_n = I R_n$, and for the equivalent single resistance replacing the combination, $V_{\text{total}} = I R_{\text{total}}$.

<!DOCTYPE html>
<html lang="en">
<head>
  <meta charset="UTF-8">
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  <meta name="aspect-ratio" content="4:3">
  <link rel="stylesheet" href="{{VENDOR_CDN}}/katex.min.css">
  <script src="{{VENDOR_CDN}}/katex.min.js"></script>
  <style>
    * { box-sizing: border-box; margin: 0; padding: 0; }
    body {
      font-family: system-ui, -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, sans-serif;
      background: #ffffff;
      color: #1e2945;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: space-between;
      min-height: 100%;
      padding: 12px;
      gap: 10px;
    }

    .circuit-box {
      width: 100%;
      max-width: 540px;
      background: #f8fafc;
      border: 1px solid #e2e8f0;
      border-radius: 12px;
      padding: 8px 12px;
      display: flex;
      flex-direction: column;
      align-items: center;
      position: relative;
    }

    svg {
      width: 100%;
      height: auto;
      max-height: 140px;
      overflow: visible;
    }

    .math-card {
      width: 100%;
      max-width: 540px;
      background: #ffffff;
      border: 1.5px solid #e0e7ff;
      border-radius: 12px;
      padding: 12px 14px;
      min-height: 100px;
      display: flex;
      flex-direction: column;
      justify-content: center;
      align-items: center;
      box-shadow: 0 2px 8px rgba(79, 70, 229, 0.05);
      position: relative;
      transition: all 0.3s ease;
    }

    .step-tag {
      font-size: 11px;
      font-weight: 700;
      text-transform: uppercase;
      letter-spacing: 0.05em;
      color: #4f46e5;
      background: #eef2ff;
      padding: 2px 8px;
      border-radius: 6px;
      margin-bottom: 6px;
    }

    .math-display {
      font-size: 19px;
      color: #1e2945;
      display: flex;
      align-items: center;
      justify-content: center;
      min-height: 38px;
      text-align: center;
    }

    .controls {
      display: flex;
      align-items: center;
      justify-content: center;
      gap: 8px;
      width: 100%;
      max-width: 540px;
    }

    .btn {
      background: #f1f5f9;
      border: 1px solid #cbd5e1;
      color: #334155;
      font-size: 13px;
      font-weight: 600;
      padding: 6px 14px;
      border-radius: 8px;
      cursor: pointer;
      display: inline-flex;
      align-items: center;
      justify-content: center;
      transition: background 0.2s, border-color 0.2s;
      user-select: none;
    }
    .btn:active {
      background: #e2e8f0;
    }
    .btn-primary {
      background: #4f46e5;
      border-color: #4f46e5;
      color: #ffffff;
    }
    .btn-primary:active {
      background: #4338ca;
    }

    .step-pill-group {
      display: flex;
      gap: 4px;
    }

    .step-pill {
      width: 28px;
      height: 28px;
      border-radius: 6px;
      background: #f1f5f9;
      border: 1px solid #e2e8f0;
      color: #64748b;
      font-size: 12px;
      font-weight: 700;
      display: flex;
      align-items: center;
      justify-content: center;
      cursor: pointer;
      transition: all 0.2s;
    }
    .step-pill.active {
      background: #4f46e5;
      color: #ffffff;
      border-color: #4f46e5;
      box-shadow: 0 0 0 2px rgba(79, 70, 229, 0.2);
    }

    /* SVG Diagram Styles */
    .wire { stroke: #64748b; stroke-width: 2.5; fill: none; stroke-linecap: round; }
    .resistor { stroke: #1e2945; stroke-width: 2.5; fill: #ffffff; }
    .resistor.hl { stroke: #4f46e5; stroke-width: 3.5; fill: #eef2ff; }
    .label-v { font-size: 13px; fill: #d97706; font-weight: 700; font-family: inherit; }
    .label-r { font-size: 14px; fill: #1e2945; font-weight: 700; font-family: inherit; }
    .label-i { font-size: 13px; fill: #0284c7; font-weight: 700; font-family: inherit; }
    .current-arrow { fill: #0284c7; }
    .dim { opacity: 0.35; transition: opacity 0.3s; }
  </style>
</head>
<body>

  
  <div class="circuit-box">
    <svg viewBox="0 0 460 135">
      
      <path d="M 40,85 L 40,30 L 420,30 L 420,85 L 40,85" class="wire" />

      
      <g id="currentFlow">
        <path d="M 60,20 L 100,20" stroke="#0284c7" stroke-width="2" stroke-linecap="round" />
        <polygon points="104,20 96,16 96,24" class="current-arrow" />
        <text x="80" y="14" text-anchor="middle" class="label-i">I (same)</text>
      </g>

      
      <rect id="res1" x="70" y="18" width="65" height="24" rx="4" class="resistor" />
      <text x="102.5" y="34.5" text-anchor="middle" dominant-baseline="middle" class="label-r">R₁</text>
      <text id="txtV1" x="102.5" y="58" text-anchor="middle" class="label-v">V₁</text>

      
      <rect id="res2" x="180" y="18" width="65" height="24" rx="4" class="resistor" />
      <text x="212.5" y="34.5" text-anchor="middle" dominant-baseline="middle" class="label-r">R₂</text>
      <text id="txtV2" x="212.5" y="58" text-anchor="middle" class="label-v">V₂</text>

      
      <rect id="res3" x="290" y="18" width="65" height="24" rx="4" class="resistor" />
      <text x="322.5" y="34.5" text-anchor="middle" dominant-baseline="middle" class="label-r">R₃</text>
      <text id="txtV3" x="322.5" y="58" text-anchor="middle" class="label-v">V₃</text>

      
      <line x1="210" y1="75" x2="210" y2="95" stroke="#1e2945" stroke-width="3" />
      <line x1="222" y1="80" x2="222" y2="90" stroke="#1e2945" stroke-width="2" />
      <text id="txtVtot" x="216" y="116" text-anchor="middle" class="label-v">V_total</text>
    </svg>
  </div>

  
  <div class="math-card" id="mathCard">
    <div class="step-tag" id="stepTag">Step 1</div>
    <div class="math-display" id="mathDisplay"></div>
  </div>

  
  <div class="controls">
    <button class="btn" id="prevBtn" aria-label="Previous">◀</button>
    <div class="step-pill-group">
      <button class="step-pill active" data-step="0">1</button>
      <button class="step-pill" data-step="1">2</button>
      <button class="step-pill" data-step="2">3</button>
      <button class="step-pill" data-step="3">4</button>
    </div>
    <button class="btn" id="nextBtn" aria-label="Next">▶</button>
    <button class="btn" id="playBtn" style="margin-left: 6px;">⏸</button>
  </div>

  <script>
    const steps = [
      {
        tag: "1. Total Voltage (Sum of Drops)",
        tex: "V_{\\text{total}} = V_1 + V_2 + V_3",
        hl: ["txtVtot", "txtV1", "txtV2", "txtV3"]
      },
      {
        tag: "2. Apply Ohm's Law (V = I · R)",
        tex: "{\\color{#0284c7}I} R_{\\text{total}} = {\\color{#0284c7}I} R_1 + {\\color{#0284c7}I} R_2 + {\\color{#0284c7}I} R_3",
        hl: ["res1", "res2", "res3", "currentFlow"]
      },
      {
        tag: "3. Factor Out Common Current",
        tex: "{\\color{#0284c7}I} R_{\\text{total}} = {\\color{#0284c7}I} (R_1 + R_2 + R_3)",
        hl: ["currentFlow"]
      },
      {
        tag: "4. Divide Both Sides by I",
        tex: "{\\mathbf{R_{\\text{total}} = R_1 + R_2 + R_3}}",
        hl: ["res1", "res2", "res3"]
      }
    ];

    let current = 0;
    let playing = true;
    let timer = null;

    const mathDisplay = document.getElementById("mathDisplay");
    const stepTag = document.getElementById("stepTag");
    const pills = document.querySelectorAll(".step-pill");
    const playBtn = document.getElementById("playBtn");
    const prevBtn = document.getElementById("prevBtn");
    const nextBtn = document.getElementById("nextBtn");

    function renderMath(tex, el) {
      if (window.katex) {
        katex.render(tex, el, { throwOnError: false });
      } else {
        el.textContent = tex;
      }
    }

    function update() {
      const s = steps[current];
      stepTag.textContent = s.tag;
      renderMath(s.tex, mathDisplay);

      pills.forEach((p, i) => {
        p.classList.toggle("active", i === current);
      });

      // SVG Highlights
      const allHl = ["res1", "res2", "res3", "txtV1", "txtV2", "txtV3", "txtVtot", "currentFlow"];
      allHl.forEach(id => {
        const el = document.getElementById(id);
        if (!el) return;
        if (s.hl.includes(id)) {
          el.classList.remove("dim");
          if (el.tagName.toLowerCase() === "rect") el.classList.add("hl");
        } else {
          el.classList.add("dim");
          if (el.tagName.toLowerCase() === "rect") el.classList.remove("hl");
        }
      });
    }

    function setStep(idx) {
      current = (idx + steps.length) % steps.length;
      update();
    }

    function next() { setStep(current + 1); }
    function prev() { setStep(current - 1); }

    function startAuto() {
      stopAuto();
      timer = setInterval(next, 3500);
      playing = true;
      playBtn.textContent = "⏸";
    }

    function stopAuto() {
      if (timer) clearInterval(timer);
      playing = false;
      playBtn.textContent = "▶";
    }

    playBtn.onclick = () => {
      if (playing) stopAuto();
      else startAuto();
    };

    prevBtn.onclick = () => { stopAuto(); prev(); };
    nextBtn.onclick = () => { stopAuto(); next(); };

    pills.forEach((p, idx) => {
      p.onclick = () => {
        stopAuto();
        setStep(idx);
      };
    });

    // Render static V_total label in SVG via KaTeX or text
    update();
    startAuto();
  </script>
</body>
</html>

Dividing both sides of the equation by the non-zero common current $I$ yields the general **series equivalent resistance** formula:

$$R_{\text{total}} = R_1 + R_2 + \dots + R_n = \sum_{i=1}^{n} R_i$$

How does this look with real numerical values in a DC circuit?

Consider a $24\text{ V}$ battery connected across three series resistors: $R_1 = 3\,\Omega$, $R_2 = 5\,\Omega$, and $R_3 = 4\,\Omega$.

First, sum the resistances to find the **equivalent resistance**: $$R_{\text{total}} = 3\,\Omega + 5\,\Omega + 4\,\Omega = 12\,\Omega$$

Next, compute the circuit current using $I = \frac{V_{\text{total}}}{R_{\text{total}}} = \frac{24\text{ V}}{12\,\Omega} = 2\text{ A}$, which gives individual drops of $V_1 = 2\text{ A} \times 3\,\Omega = 6\text{ V}$, $V_2 = 10\text{ V}$, and $V_3 = 8\text{ V}$.

<!DOCTYPE html>
<html lang="en">
<head>
  <meta charset="UTF-8">
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  <meta name="aspect-ratio" content="4:3">
  <style>
    * { box-sizing: border-box; margin: 0; padding: 0; }
    body {
      font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, sans-serif;
      background: #f8f9fa;
      color: #1e2945;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: center;
      min-height: 100%;
      padding: 12px;
      user-select: none;
      -webkit-user-select: none;
    }

    .main-wrap {
      width: 100%;
      max-width: 520px;
      display: flex;
      flex-direction: column;
      gap: 12px;
    }

    /* Circuit SVG styling */
    .circuit-box {
      background: #ffffff;
      border-radius: 12px;
      padding: 10px 14px 4px;
      box-shadow: 0 2px 8px rgba(0,0,0,0.04);
      border: 1px solid #e2e8f0;
      position: relative;
    }

    svg {
      width: 100%;
      height: auto;
      display: block;
      overflow: visible;
    }

    .wire {
      stroke: #94a3b8;
      stroke-width: 3;
      fill: none;
      stroke-linecap: round;
      stroke-linejoin: round;
    }

    .resistor-box {
      cursor: pointer;
      transition: filter 0.2s ease, transform 0.2s ease;
    }
    .resistor-box:hover {
      filter: brightness(0.96);
    }

    .current-dot {
      fill: #10b981;
    }

    /* Voltage Bar Section */
    .bar-card {
      background: #ffffff;
      border-radius: 12px;
      padding: 12px 14px;
      box-shadow: 0 2px 8px rgba(0,0,0,0.04);
      border: 1px solid #e2e8f0;
      display: flex;
      flex-direction: column;
      gap: 8px;
    }

    .bar-track {
      display: flex;
      width: 100%;
      height: 38px;
      border-radius: 8px;
      overflow: hidden;
      border: 1px solid #cbd5e1;
      background: #f1f5f9;
    }

    .segment {
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: center;
      color: #ffffff;
      font-weight: 700;
      font-size: 13px;
      cursor: pointer;
      transition: opacity 0.2s ease;
    }

    .seg-1 { background: #3b82f6; width: 25%; }
    .seg-2 { background: #10b981; width: 41.66%; }
    .seg-3 { background: #f59e0b; width: 33.34%; }

    .seg-dim { opacity: 0.35; }

    .math-row {
      display: flex;
      justify-content: space-between;
      align-items: center;
      font-size: 13px;
      font-weight: 600;
      color: #64748b;
      padding: 0 4px;
    }

    .badge {
      display: inline-flex;
      align-items: center;
      gap: 4px;
      padding: 3px 8px;
      border-radius: 6px;
      font-size: 12px;
      font-weight: 700;
    }

    .badge-blue { background: #eff6ff; color: #2563eb; }
    .badge-green { background: #ecfdf5; color: #059669; }
    .badge-amber { background: #fffbeb; color: #d97706; }
    .badge-total { background: #1e2945; color: #ffffff; }

    .interactive-hint {
      text-align: center;
      font-size: 12px;
      font-weight: 600;
      color: #94a3b8;
    }
  </style>
</head>
<body>

<div class="main-wrap">
  
  <div class="circuit-box">
    <svg viewBox="0 0 420 180">
      
      <path id="circuitPath" class="wire" d="M 50,90 L 50,30 L 370,30 L 370,150 L 50,150 L 50,90" />

      
      <g stroke="#10b981" stroke-width="2" fill="none" stroke-linecap="round">
        <path d="M 125,27 L 132,30 L 125,33" />
        <path d="M 373,85 L 370,92 L 367,85" />
        <path d="M 225,147 L 218,150 L 225,153" />
      </g>
      <text x="145" y="22" font-size="12" font-weight="700" fill="#10b981">I = 2 A</text>

      
      <g transform="translate(50, 90)">
        <line x1="-16" y1="-12" x2="16" y2="-12" stroke="#1e2945" stroke-width="4" stroke-linecap="round"/>
        <line x1="-8" y1="-2" x2="8" y2="-2" stroke="#64748b" stroke-width="2.5" stroke-linecap="round"/>
        <line x1="-16" y1="8" x2="16" y2="8" stroke="#1e2945" stroke-width="4" stroke-linecap="round"/>
        <line x1="-8" y1="18" x2="8" y2="18" stroke="#64748b" stroke-width="2.5" stroke-linecap="round"/>
        <text x="-38" y="5" font-size="13" font-weight="800" fill="#1e2945">24 V</text>
        <text x="16" y="-14" font-size="11" font-weight="700" fill="#10b981">+</text>
        <text x="16" y="24" font-size="11" font-weight="700" fill="#64748b">−</text>
      </g>

      
      <g id="r1-group" class="resistor-box" transform="translate(160, 15)" onclick="selectResistor(0)">
        <rect x="0" y="0" width="60" height="30" rx="6" fill="#ffffff" stroke="#3b82f6" stroke-width="2.5"/>
        <text x="30" y="15" text-anchor="middle" dominant-baseline="middle" font-size="12" font-weight="700" fill="#1e2945">3 Ω</text>
        <text x="30" y="-7" text-anchor="middle" font-size="11" font-weight="700" fill="#2563eb">V₁ = 6 V</text>
      </g>

      
      <g id="r2-group" class="resistor-box" transform="translate(260, 15)" onclick="selectResistor(1)">
        <rect x="0" y="0" width="60" height="30" rx="6" fill="#ffffff" stroke="#10b981" stroke-width="2.5"/>
        <text x="30" y="15" text-anchor="middle" dominant-baseline="middle" font-size="12" font-weight="700" fill="#1e2945">5 Ω</text>
        <text x="30" y="-7" text-anchor="middle" font-size="11" font-weight="700" fill="#059669">V₂ = 10 V</text>
      </g>

      
      <g id="r3-group" class="resistor-box" transform="translate(210, 135)" onclick="selectResistor(2)">
        <rect x="0" y="0" width="60" height="30" rx="6" fill="#ffffff" stroke="#f59e0b" stroke-width="2.5"/>
        <text x="30" y="15" text-anchor="middle" dominant-baseline="middle" font-size="12" font-weight="700" fill="#1e2945">4 Ω</text>
        <text x="30" y="42" text-anchor="middle" font-size="11" font-weight="700" fill="#d97706">V₃ = 8 V</text>
      </g>

      
      <circle id="chargeDot" class="current-dot" r="4.5" cx="50" cy="30" />
    </svg>
  </div>

  
  <div class="bar-card">
    <div class="bar-track">
      <div id="bar-0" class="segment seg-1" onclick="selectResistor(0)">6 V</div>
      <div id="bar-1" class="segment seg-2" onclick="selectResistor(1)">10 V</div>
      <div id="bar-2" class="segment seg-3" onclick="selectResistor(2)">8 V</div>
    </div>

    <div class="math-row">
      <div>
        <span class="badge badge-blue">6 V</span> +
        <span class="badge badge-green">10 V</span> +
        <span class="badge badge-amber">8 V</span>
      </div>
      <div>
        = <span class="badge badge-total">24 V</span>
      </div>
    </div>
  </div>

  <div class="interactive-hint">Tap any resistor to inspect drop: V = I × R</div>
</div>

<script>
  let activeIndex = -1;
  const groups = [
    document.getElementById('r1-group'),
    document.getElementById('r2-group'),
    document.getElementById('r3-group')
  ];
  const bars = [
    document.getElementById('bar-0'),
    document.getElementById('bar-1'),
    document.getElementById('bar-2')
  ];

  function selectResistor(idx) {
    if (activeIndex === idx) {
      activeIndex = -1;
    } else {
      activeIndex = idx;
    }
    updateDisplay();
  }

  function updateDisplay() {
    groups.forEach((g, i) => {
      const rect = g.querySelector('rect');
      if (activeIndex === -1 || activeIndex === i) {
        g.style.opacity = '1';
        rect.style.strokeWidth = (activeIndex === i) ? '3.5' : '2.5';
      } else {
        g.style.opacity = '0.35';
        rect.style.strokeWidth = '2.5';
      }
    });

    bars.forEach((b, i) => {
      if (activeIndex === -1 || activeIndex === i) {
        b.classList.remove('seg-dim');
      } else {
        b.classList.add('seg-dim');
      }
    });
  }

  // Smooth continuous circuit animation
  const dot = document.getElementById('chargeDot');
  const path = document.getElementById('circuitPath');
  const pathLength = path.getTotalLength();
  let progress = 0;

  function animateCurrent() {
    progress = (progress + 1.2) % pathLength;
    const pt = path.getPointAtLength(progress);
    dot.setAttribute('cx', pt.x);
    dot.setAttribute('cy', pt.y);
    requestAnimationFrame(animateCurrent);
  }
  requestAnimationFrame(animateCurrent);
</script>

</body>
</html>

Notice how the sum of the voltages ($6\text{ V} + 10\text{ V} + 8\text{ V} = 24\text{ V}$) perfectly matches the battery source voltage.

Why do many beginners incorrectly predict that the last resistor receives less current than the first?

## Common Misconception: Current Does Not Get "Used Up"

A common mistake is believing that the first resistor consumes current, leaving less for subsequent resistors down the line.

Current is the rate of flow of electric charges, and charge cannot disappear; charges lose **electrical potential energy** (voltage), while the quantity of charge flowing past any point per second ($I$) remains identical throughout the entire chain.

electronics

Deriving the Series Equivalent Resistance Formula. Derive the series equivalent resistance relationship R_total = R1 + R2 + ... + Rn using potential difference summation.

Resistors in Series & Parallel: Formal Derivations & Circuit Rules

Resistors in Series: Conceptual Analysis and Mathematical Derivation

Deriving the Equation for Resistors in Series (A-Level Physics Derivations)

Pedagogical strategy focuses on rooting circuit algebraic formulas in fundamental physics conservation laws rather than procedural computation. Students first establish charge conservation ($I$ is constant) and energy conservation ($V_{\text{total}} = \sum V_i$) before applying Ohm's law substitutions. This explicit derivation pathway builds the conceptual bridge to potential divider circuits and complex DC network analysis.

Deriving the Series Equivalent Resistance Formula

The Two Core Principles

Step-by-Step Derivation