Question

Suppose there are four identical conducting spheres: A, B, C, and D. In each scenario, sphere A starts with a charge of −?,sphere C starts out with a charge of +Q, and spheres B and D start out neutral. Then, the spheres are touched to each other and separated in the order described. Any spheres not in contact are held very far away. For each scenario, give the final charge of sphere C as a whole‑number fraction of ?. For example, if the final charge of sphere C is −?/7, enter "−1" in the numerator and "7" in the denominator.

Remember, before each scenario begins, spheres A, B, C, and D are reset to charges of −?, 0, +?, and 0 again.

Scenario 1: A and B touch and are separated, then A and C touch and are separated, then A and D touch and are separated.

?C=?×

_____

Scenario 2: A and B touch and are separated, then B and D touch and are separated, then C and D touch and are separated.

?C=?×

______

Scenario 3: A and D touch and are separated, then D and C touch and are separated, then D and A touch and are separated, then A and B touch and are separated.

?C=?×

______

Answer 1

When two identical charged spheres are touched, the total charge on the two spheres gets equally distributed.

1. After A and B are touched and separated, they each have (-Q+0)/2 = -Q/2 charge. Then A and C are touched and separated, so they each have (Q - Q/2)/2 = Q/4 charge. Then A and D are touched and separated , so they each have (Q/4+0)/2 = Q/8 charge.

Final charges : A (Q/8), B (-Q/2), C (Q/4), D (Q/8)

2. After A and B are touched and separated, they each have (-Q+0)/2 = -Q/2 charge. Then B and D are touched and separate, so they each have (-Q/2+ 0)/2 = - Q/4 charge. Then C and D are touched and separated, so they each have (Q-Q/4)/2 = 3Q/8 charge.

Final charges : A (-Q/2), B (-Q/4), C (3Q/8), D (3Q/8)

3. After A and D are touched and separated, they each have (-Q+0)/2 = - Q/2 charge. Then D and C are touched and separated, so they each have (-Q/2+Q)/2 = Q/4 charge. Then D and A are touched dand separated, so they each have (Q/4-Q/2)/2 = - Q/8 charge. Then A and B are touched, so they each have (-Q/8+0)/2 = - Q/16 charge.

Final charges : A (-Q/16), B (-Q/16), C (Q/4), D (-Q/8)

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