# Not understanding work stored in a capacitor

I know V = q/c and W = Vq and dW = V dq. But why is Work in charging a capacitor W = integral of q/c dq?

q seems to represent a charge on the capacitor plate and dq seems to represent a separate test charge. If I add a charge to the capacitor plate, I take take the resulting votage and multiply by test charge dq to get the work done on dq. For each additional charge added to the capacitor, I continue to multiply the new voltage by dq. But why should I multiply by dq? The voltage created affects any amount of charge, so why is it restricted to affecting only dq?

For the basic Energy equation E = integral of F(x)dx, this means I am splitting up distance x into pieces of size dx, calculating the force at each segment, multiplying it by the distance dx over which the force acts, and adding everything up. I can see why F(x) is related to dx because each value of F(x) only works across distance dx.

## Answers and Replies

gleem
The work charging a capacitor is similar at least mathematically to the work done in compressing a spring and therefore storing energy in the spring.

for a spring dw = k⋅x⋅dx where x is the present amount of compression and k is the spring constant basically how much the spring resists compression per unit of compression.

So W = k ∫xdx = ½kx2 just as work stored in a capacitor as energy is ½Cq2

Dullard and berkeman
Yes, that's where I am getting caught up. For a spring, each small distance you move affects the spring force, and that force is applied specifically for that small distance dx.

But for a capacitor, each small charge you add to the plate increases the voltage, but why is that voltage applied for only a small test charge dq? Why would that test charge dq affect the value of the next charge that arrives at the capacitor plate?

gleem
But for a capacitor, each small charge you add to the plate increases the voltage, but why is that voltage applied for only a small test charge dq? Why would that test charge dq affect the value of the next charge that arrives at the capacitor plate?

In the spring each increment in the compression increases the resisting force that occurs for that particular small increment. Each succeeding increment results in a stronger resisting force. Similarly in the capacitor each incremental increase in charge for a capacitor increases the voltage and therefore the electric field that makes it harder for the next incremental increase in charge to be placed in the capacitor.

gleem
Ok, so dq represents the incremental charge to be applied on the capacitor plate and not an imaginary test charge between the plates?

Yes. I do not know where you got the imaginary test charge from or how it is used to describe this process.

vanhees71
$$W=\int_0^{Q} \mathrm{d} q \frac{q}{C} = \frac{Q^2}{2C}=\frac{C}{2} U^2.$$