Answer to Question #97400 in Mechanics | Relativity for Ali hamza

Question

(a) Twelve equal charges, q, are situated at the corners of a regular 12–sided polygon (for in-
stance, 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
(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.

Answer 1

We need to find the force on a test charge Q at the center and force on Q in different situations

Answer:

(a)

We know, Electric force between two charges is directly proportional to the product of those two charges and inversely proportional to the square of the distance between two charges.

$$F_e ∝ \frac { qQ}{r^2}\\ F_e = \frac {1}{4\pi \epsilon _o} \frac { qQ}{r^2}$$

Here r is the distance between the centre of the 12-sided polygon and its corner.

q and Q are the charges and $\epsilon_o$ is a constant

If we observe the diagram, we can easily understood, at the corners we have the equal charge with the same distance from the centre of the polygon. So, both of these charges can apply the equal force on the test charge in the opposite directions.

So, sum of the total forces at test charge will be zero.

(b).

One of the charge in 12 q's (means the one at “6 o’clock”) is removed, the force on the test charge at the centre is only one which is Opposite to “6 o’clock”, that is “12 o’clock”. We can write that force as

$$F_e = \frac {1}{4\pi \epsilon _o} \frac { qQ}{r^2}$$

(c). Now 13 equal charges, q, are placed at the corners of a regular 13–sided polygon.

Now, we can find the force on a test charge Q at the center

Here, also the force on the charge at the centre is zero. Because symmetry of the system.

d).

If one of the 13 q’s is removed from any corner of the Polygon, we can write a equation to net force due to the remaining 12 charges as

$$F_e = \frac {1}{4\pi \epsilon _o} \frac { qQ}{r^2}$$