Derivation for E = V/d? (capacitors)

[plazprestige]

One of the formulas I came across while doing problems with simple parallel plate capacitors was E = V/d, where E is the magnitude of the electric field between the plates, V is the potential difference between the plates, and d is the separation of the plates. I'm wondering where this formula is derived from.

I know that the electric field between the two plates of a capacitor is constant (except near the edges), but am not sure how that would play into the explanation.

 

[olivermsun]

Think of the definitions of electric field and electric potential, and then think of W = F d.

 

[plazprestige]

well the definition of an electric field is F/q, where q is in the field, and the definition of electric potential is electrical potential energy divided by charge.

E = F/q
V = E/q
W = Fd

So by substitution, W = Eqd

I want to get to E = V/d so I'll solve for E...

E = W/qd

So W/q is somehow equal to V? So W/q = E/q

And by the work energy theorem, W = delta E, and the voltage in E = V/d is in fact a potential difference.

Thanks! I literally reasoned that out while typing. Thanks for pointing me in the right direction.

 

[jtbell]

well the definition of an electric field is F/q, where q is in the field, and the definition of electric potential is electrical potential energy divided by charge.

[...snippety snip...]

So W/q is somehow equal to V?

Like you said at the beginning of your post...

 

[sardar]

This formula can be derived from the definition of electric field and the concept of capacitance. The electric field is defined as the force per unit charge experienced by a test charge placed in the field. In the case of a parallel plate capacitor, the electric field is constant and directed from the positive plate to the negative plate.

The potential difference, V, between the plates is defined as the work done per unit charge in moving a test charge from one plate to the other. This can also be expressed as the product of the electric field and the distance between the plates, or V = Ed.

The capacitance, C, of a parallel plate capacitor is defined as the ratio of the charge on the plates to the potential difference between them, or C = Q/V.

Combining these equations, we get E = V/d = Q/dC. This shows that the electric field is directly proportional to the charge on the plates and inversely proportional to the distance between them. This relationship is also known as Coulomb's Law.

In summary, the formula E = V/d for a parallel plate capacitor is derived from the definitions of electric field, potential difference, and capacitance. It shows the relationship between these quantities and helps us understand the behavior of capacitors in electrical circuits.