3) A thin ring made of uniformly charged insulating material has total charge Q

Question

3) A thin ring made of uniformly charged insulating material has total charge Q and radius R. The ring is positioned along the x-y plane of a 3d coordinate system such that the center of the ring is at the origin of the coordinate system. (a) Determine an expression for the potential at an arbitrary location along the z-axis in terms of Q, R, and z. (b) Use this expression to determine an expression for the magnitude of the electric field at an arbitrary location along the z-axis in terms of Q, R, and z. Hint: Apply the technique of charge integration in part (a). 3) A thin ring made of uniformly charged insulating material has total charge Q and radius R. The ring is positioned along the x-y plane of a 3d coordinate system such that the center of the ring is at the origin of the coordinate system. (a) Determine an expression for the potential at an arbitrary location along the z-axis in terms of Q, R, and z. (b) Use this expression to determine an expression for the magnitude of the electric field at an arbitrary location along the z-axis in terms of Q, R, and z. Hint: Apply the technique of charge integration in part (a).

Explanation / Answer

3)

a) distance from each point on the ring to the point on +z axis at a distance z from origin,

r = sqrt(R^2 + z^2)

potential due to a small segment of charge dQ,

dV = k*dQ/r

= k*dQ/sqrt(R^2 + z^2)

so, potential due to the whole ring at the given point,

V = integral dV

= integral k*dQ/r

= k*Q/r

= k*Q/sqrt(R^2 + z^2) <<<<<<<<<<<<<-----------Answer

b) we know, E = -dV/dz

= -k*Q*(-1/2)*(R^2 + z^2)^(-1/2 - 1)*(2*z)

= k*Q*z/(R^2 + z^2)^(3/2) <<<<<<<<<<<<<-----------Answer

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