Derive an Expression for Energy Stored in a Capacitor
The energy stored in a capacitor is given by the equation.
E = (1/2) * C * V^2
Where: E is the energy stored in the capacitor, C is the capacitance of the capacitor, and V is the voltage across the capacitor.
This formula represents the energy stored in an ideal capacitor without considering factors like dielectric losses or resistance.
Derivation
The energy stored in a capacitor can be derived using calculus and the concept of work.
Consider a capacitor with a charge q on its plates and a voltage difference V across it. The capacitance C of the capacitor is defined as the ratio of the charge on the plates to the voltage across them:
C = q / V
To calculate the work done in moving a small increment of charge dq from one plate to the other against the electric field, we can use the formula for work:
dW = V * dq
Here, dW represents the work done and dq is a small increment of charge. Since the voltage V remains constant during the process, we can rearrange the equation to isolate dq:
dq = C * dV
Now, we integrate both sides of the equation to find the total charge q:
∫dq = ∫C * dV q = ∫C * dV
Next, we substitute the expression for C from the initial definition:
q = ∫(q / V) * dV
We can simplify this equation by multiplying both sides by V:
q * V = ∫q * dV
Now, we integrate both sides of the equation:
∫(q * V) = ∫q * dV
The left-hand side of the equation represents the energy stored in the capacitor, denoted as E. The right-hand side is the integral of q with respect to V, which can be written as:
E = ∫q * dV
To solve this integral, we express q as a function of V. Since q = C * V, we substitute this in:
E = ∫(C * V) * dV
Now we can integrate the right-hand side:
E = C * ∫V * dV E = C * (V^2 / 2)
Finally, substituting the expression for C = q / V, we obtain the formula for the energy stored in a capacitor:
E = (1/2) * C * V^2
That’s how the formula for energy stored in a capacitor is derived.
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