Answer to Question #212833 in Electricity and Magnetism for NICKO

Question #212833
A insulating rod is bent into a semicircular arc of radius a, and a total charge Q is distributed uniformly along the rod. Calculate the electric potential at the center of curvature of the arc if the potential is assumed to be zero at infinity.
1
Expert's answer
2021-07-05T17:53:34-0400

Radius=a

Charge=Q

V=dq4πϵ0rV=\smallint \frac{dq}{4 \pi\epsilon_0 r}

dq=λdldq=\lambda dl

dl=adθdl=ad\theta

λ=θπa\lambda=\frac{\theta}{\pi a}

dq=θπa(adθ)dq=\frac{\theta}{\pi a} (ad\theta)

dq=θdθπdq=\frac{\theta d\theta}{\pi}



Limit

θ=0°\theta=0° to θ=π\theta=\pi

V=14πϵ0dqaV=\frac{1}{4\pi\epsilon_0}\int\frac{dq}{a}

V=14πϵ00πQdθπaV=\frac{1}{4\pi\epsilon_0}\smallint _{0}^ \pi \frac{Q d\theta}{\pi a}

V=Q4πϵ0aV=\frac{Q}{4 \pi\epsilon_0 a}


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