Questions: Energy Storage in Capacitors and Inductors

Question 1

A capacitor is charged to 10V and connected in a circuit via a switch. At t=0, the switch closes. What is the capacitor voltage at t=0⁺ (immediately after closing)?

• A. 0V — the circuit demands current flow, which instantly redistributes the charge
• B. 10V — capacitor voltage cannot change instantaneously because that would require infinite power
• C. 5V — voltage splits evenly between the capacitor and the rest of the circuit
• D. It depends on the resistance in the circuit — with zero resistance, voltage drops to zero

Answer: (B)

Capacitor voltage cannot change instantaneously. The energy stored is W_C = ½CV², and changing voltage instantly would require instantaneous energy transfer, which demands infinite power (P = dW/dt). Therefore, the capacitor voltage at t=0⁺ equals the voltage at t=0⁻ = 10V. This is the initial condition for transient analysis. Option D contains a subtlety: with zero resistance, you get a theoretical contradiction — the math breaks down, but physically it just means the transition is extremely fast, not truly instantaneous. In practice, all real circuits have some resistance.

Question 2

Which statement correctly captures the duality between capacitors and inductors with respect to their continuity constraints?

• A. Both capacitor voltage and inductor current can change instantaneously if the applied voltage or current is large enough
• B. Capacitor voltage cannot change instantaneously; inductor current cannot change instantaneously — both because stored energy cannot change instantaneously
• C. Capacitor current cannot change instantaneously; inductor voltage cannot change instantaneously
• D. Capacitors resist current changes; inductors resist voltage changes

Answer: (B)

The continuity constraints follow directly from energy storage: W_C = ½CV² means voltage cannot jump (that would require infinite current through C = i/(dV/dt)); W_L = ½LI² means current cannot jump (that would require infinite voltage across L = V/(dI/dt)). Option C swaps which quantity is continuous — current through a capacitor CAN change instantly, and voltage across an inductor CAN change instantly; it's voltage across C and current through L that are continuous. Option D inverts the mapping between element and quantity.

Question 3

When a switch suddenly opens in a series circuit containing an inductor carrying 2A, the current through the inductor immediately drops to zero.

• T. True
• F. False

Answer: False

Inductor current cannot change instantaneously because W_L = ½LI² — instantaneous change requires infinite power. When the switch opens, the inductor 'fights' to maintain its current. With no path for current to flow through the switch, the inductor generates a very large voltage spike (sometimes thousands of volts) to drive current through whatever path exists — this is why opening inductive circuits without a protection diode or snubber can destroy switches and components. The current does eventually decay to zero, but through a transient process, not instantaneously.

Question 4

The energy stored in a capacitor increases quadratically with voltage: doubling the voltage quadruples the stored energy.

• T. True
• F. False

Answer: True

W_C = ½CV², so doubling V (V → 2V) gives W = ½C(2V)² = 4 × ½CV². The stored energy quadruples. This quadratic relationship arises because to push more charge onto the capacitor, you must work against the increasing electric field created by the charge already there — the work required per additional charge scales with the voltage, which itself scales with charge. The same quadratic appears in W_L = ½LI² for inductors and in mechanical energy storage: W_spring = ½kx² and W_kinetic = ½mv², all following from integrating a force (or voltage) that itself grows with displacement (or current).

Question 5

Why can't a capacitor's voltage change instantaneously? Explain using energy storage principles rather than just citing the formula i = C·dv/dt.

Model answer: A capacitor stores energy W_C = ½CV² in its electric field. If the voltage were to change instantaneously — say from V₁ to V₂ in zero time — the stored energy would change from ½CV₁² to ½CV₂² in zero time. Power is P = dW/dt; dividing a finite energy change by zero time gives infinite power. No physical circuit can supply or absorb infinite power, so instantaneous voltage change is physically impossible. The circuit's past history (what voltage was stored before t=0) therefore completely determines the initial condition for any transient that follows — the capacitor voltage at t=0⁺ always equals the voltage at t=0⁻.

The energy-based argument is more fundamental than the formula i = C·dv/dt. The formula says instantaneous voltage change requires infinite current, which is true — but the deeper reason is that you cannot transfer infinite power. Understanding this connects the circuit constraint to a universal physical principle: conservation of energy and the finite rate at which energy can be transferred. The same argument applies to inductors with current: instantaneous current change would require infinite power into the magnetic field.