physics

Two capacitor paradox

Two capacitor paradox is a physics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Two capacitor paradox rather than just read about it. In short: The two capacitor paradox or capacitor paradox is a paradox, or counterintuitive thought experiment, in electric circuit theory. The thought experiment is usually described as follows: Two identical capacitors are connected in parallel with an open switch between them.

Key takeaways

Two capacitor paradox belongs to physics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Two capacitor paradox to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Two capacitor paradox from memory before moving on to harder problems.

Reference excerpt

The two capacitor paradox or capacitor paradox is a paradox, or counterintuitive thought experiment, in electric circuit theory. The thought experiment is usually described as follows:

Two identical capacitors are connected in parallel with an open switch between them. One of the capacitors is charged with a voltage of V i {\displaystyle V_{i}} , the other is uncharged. When the switch is closed, some of the charge Q = C V i {\displaystyle Q=CV_{i}} on the first capacitor flows into the second, reducing the voltage on the first and increasing the voltage on the second. When a steady state is reached and the current goes to zero, the voltage on the two capacitors must be equal since they are connected together. Since they both have the same capacitance C {\displaystyle C} the charge will be divided equally between the capacitors so each capacitor will have a charge of Q 2 {\displaystyle {Q \over 2}} and a voltage of V f = Q 2 C = V i 2 {\displaystyle V_{f}={Q \over 2C}={V_{i} \over 2}} . At the beginning of the experiment the total initial energy W i {\displaystyle W_{i}} in the circuit is the energy stored in the charged capacitor:

W i = 1 2 C V i 2 {\displaystyle W_{i}={1 \over 2}CV_{i}^{2}}

At the end of the experiment the final energy W f {\displaystyle W_{f}} is equal to the sum of the energy in the two capacitors:

W f = 1 2 C V f 2 + 1 2 C V f 2 = C V f 2 = C ( V i 2 ) 2 = 1 4 C V i 2 = 1 2 W i {\displaystyle W_{f}={1 \over 2}CV_{f}^{2}+{1 \over 2}CV_{f}^{2}=CV_{f}^{2}=C\left({V_{i} \over 2}\right)^{2}={1 \over 4}CV_{i}^{2}={1 \over 2}W_{i}}

Thus the final energy W f {\displaystyle W_{f}} is equal to half of the initial energy W i {\displaystyle W_{i}} . The paradox lies in the unexplained loss of the remainder half of the initial energy: an apparent violation of the law of conservation of energy.

Solutions This problem has been discussed in electronics literature at least as far back as 1955. Unlike some other paradoxes in science, this paradox is not due to the underlying physics, but to the limitations of the 'ideal circuit' conventions used in circuit theory. The description specified above is not physically realizable if the circuit is assumed to be made of ideal circuit elements, as is usual in circuit theory. If the series resistance of the wires and conductors in the circuit is R {\displaystyle R} , the initial current when the switch is closed is

i ( 0 ) = V i R {\displaystyle i(0)={V_{\text{i}} \over R}}

If the wires connecting the two capacitors, the switch, and the capacitors themselves are idealized as having no electrical resistance or inductance as is usual, then closing the switch would connect points at different voltage with a perfect conductor, causing an infinite current to flow, which is impossible. Therefore a solution requires that one or more of the 'ideal' characteristics of the elements in the circuit be relaxed, which was not specified in the above description. The solution differs depending on which of the assumptions about the actual characteristics of the circuit elements is abandoned:

… excerpt ends here. Continue reading the full article.

Illustrations

Worked examples

Example 1 — a first encounter with Two capacitor paradox

Start with the simplest possible case. Write down what Two capacitor paradox claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In physics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Two capacitor paradox before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Two capacitor paradox ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Two capacitor paradox

In research: Two capacitor paradox appears in physics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Two capacitor paradox in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Two capacitor paradox is common in secondary-school and first-year university syllabi. It links to neighbouring topics Capacitors, Electrical circuits, Physical paradoxes, so understanding it makes those chapters shorter.
In everyday life: Look for Two capacitor paradox outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Two capacitor paradox in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Two capacitor paradox means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Two capacitor paradox out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Two capacitor paradox in simple terms?

The two capacitor paradox or capacitor paradox is a paradox, or counterintuitive thought experiment, in electric circuit theory. The thought experiment is usually described as follows: Two identical capacitors are connected in parallel with an open switch between them.

Why does Two capacitor paradox matter?

Because it connects several physics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Two capacitor paradox?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Two capacitor paradox.