(a) Twelve equal charges, q, are situated at the corners of a regular 12-sided polygon (for instance, one on each numeral of a clock face). What is the net force on a test charge Q at the center? (b) Suppose one of the 12 q’s is removed (the one at “6 o’clock”). What is the force on Q? Explain your reasoning carefully. 2.1 The Electric Field 61 (c) Now 13 equal charges, q, are placed at the corners of a regular 13-sided polygon. What is the force on a test charge Q at the center? (d) If one of the 13 q’s is removed, what is the force on Q? Explain your reasoning.

(a) The net force on a test charge Q at the center of a regular 12-sided polygon with equal charges q at each corner would be zero. This is because the electric force is a vector quantity and has both magnitude and direction. The forces from each of the charges on the test charge Q would be equal in magnitude but opposite in direction, thus they would cancel each other out.

(b) If one of the 12 charges is removed, the symmetry is broken and there will be a net force on Q. The force will be directed towards the missing charge (at "6 o'clock") because the forces from the remaining charges will no longer cancel out completely. The magnitude of this force can be calculated using Coulomb's law.

(c) Similarly, if 13 equal charges q are placed at the corners of a regular 13-sided polygon, the net force on a test charge Q at the center would still be zero. This is because the forces from each of the charges on the test charge Q would still be equal in magnitude but opposite in direction, thus they would cancel each other out.

(d) If one of the 13 charges is removed, the symmetry is again broken and there will be a net force on Q. The force will be directed towards the missing charge because the forces from the remaining charges will no longer cancel out completely. The magnitude of this force can be calculated using Coulomb's law.