A point charge $+ q$ is moved with a constant velocity $v$ toward a solid conducting sphere carrying a charge $Q$ . Find the (a) potential of the sphere, (b) charge induced as the function of $r$ , (c) charge flowing if we close the key $K$ , and (d) current in the part (c).

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Solution

a. We find potential at the center of sphere.
$V = \frac{k q}{r} + \frac{K Q}{R} = \frac{1}{4 π ε_{0}} [\frac{q}{r} + \frac{Q}{R}]$
b. If key is closed, potential of sphere should be zero.
$V = 0$ or $K [\frac{q}{r} + \frac{Q}{R}] = 0$ or $Q = - \frac{q R}{r}$
c. Charge flowing through key is $q^{'} = \frac{q R}{r}$
d. $I = \frac{d q^{'}}{d t} = \frac{q R}{- r^{2}} \frac{d r}{d t} = \frac{q R}{- r^{2}} (- v) = \frac{q R v}{r^{2}}$

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A point charge +q is moved with a constant velocity v toward a solid conducting sphere carrying a charge Q. Find the (a) potential of the sphere, (b) charge induced as the function of r, (c) charge flowing if we close the key K, and (d) current in the part (c).

a. We find potential at the center of sphere. V=kqr+KQR=14πε0[qr+QR] b. If key is closed, potential of sphere should be zero. V=0 or K[qr+QR]=0 or Q=−qRrc. Charge flowing through key is q′=qRrd. I=dq′dt=qR−r2drdt=qR−r2(−v)=qRvr2

A point charge $+ q$ is moved with a constant velocity $v$ toward a solid conducting sphere carrying a charge $Q$ . Find the (a) potential of the sphere, (b) charge induced as the function of $r$ , (c) charge flowing if we close the key $K$ , and (d) current in the part (c).