- #1

AndresPB

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- Homework Statement
- Find the Electric Potential Energy of half a circle charged with a Linear Charge Density: "Lambda" along its z axis. After that, find the Electric Field "E" deriving its potential energy.

- Relevant Equations
- Potential Energy dV = (1/(4*Pi*Epsilon_0))*(dQ/|r-r'|)

Electric Field E = - ∇ V

So I figured out the potential is: dV = (1/(4*Pi*Epsilon_0))*[λ dl/sqrt(z^2+a^2)]

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From that expression: We can figure out that since its half a ring we have to integrate from 0 to pi*a, so we would get:

V = (1/(4*Pi*Epsilon_0))*[λ {pi*a]/sqrt(z^2+a^2)]

In that expression: a = sqrt(x^2+y^2)

What I am having trouble is getting the electric field from this expression. When I do ∇ V, I am getting three components for my electric field in the (x,y,z), directions. But when I analyze the problem. It is clear to me that there is not a "y" component, as there is simetry along this axis. What shall I do?

The answer I am getting at the moment is this ("L" in this case is: (1/(4*Pi*Epsilon_0))*[λ {pi}]):

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