What is the general expression for the energy stored in a capacitor in terms its capacitance C and voltage V?

What is the general expression for the energy stored in a capacitor in terms its capacitance C and voltage V?

• A. E = 1/2 C V^2
• B. E = C V
• C. E = V^2 /(2C)
• D. E = I^2 R t

Answer: (A)

When a capacitor is charged, energy is stored in the electric field between its plates. The work needed to add a small amount of charge dq at the instantaneous voltage V is V dq, so the total stored energy is E = ∫ V dq from 0 to the final charge Q. For a capacitor, q = C V, so V = q/C. Substituting and integrating gives E = ∫_0^Q (q/C) dq = Q^2/(2C). Since Q = C V, this becomes E = (1/2) C V^2. This is why the energy stored is proportional to the capacitance and to the square of the voltage. The other expressions don’t fit: CV has units of charge, not energy; V^2/(2C) would not equal (1/2) C V^2 in general; and I^2 R t represents energy dissipated as heat in a resistor, not the energy stored in a capacitor.

When a capacitor is charged, energy is stored in the electric field between its plates. The work needed to add a small amount of charge dq at the instantaneous voltage V is V dq, so the total stored energy is E = ∫ V dq from 0 to the final charge Q. For a capacitor, q = C V, so V = q/C. Substituting and integrating gives E = ∫_0^Q (q/C) dq = Q^2/(2C). Since Q = C V, this becomes E = (1/2) C V^2. This is why the energy stored is proportional to the capacitance and to the square of the voltage.

The other expressions don’t fit: CV has units of charge, not energy; V^2/(2C) would not equal (1/2) C V^2 in general; and I^2 R t represents energy dissipated as heat in a resistor, not the energy stored in a capacitor.