Question

Two point charges $ +Q $ and $ -4Q $ are placed 60 cm apart. Where should a third charge be placed so that it experiences zero net force?

• A. 15 cm from \( +Q \)
• B. 20 cm from \( -4Q \)
• C. 60 cm from \( +Q \)
• D. 45 cm from \( -4Q \)

Hint:

Always consider direction of forces when solving electrostatics problems. For equilibrium points, test regions outside the segment first, especially when dealing with unlike charges. Use Coulomb's law and set magnitudes equal to find the exact distance.

Answer 1

The Correct Option is C

Let the charges \( +Q \) and \( -4Q \) be fixed on a line with a separation of 60 cm. We are to find a location to place a third charge (say \( q \)) such that the net electrostatic force acting on it is zero.
We consider three possible regions to place the third charge: 
(i) to the left of \( +Q \), (ii) between \( +Q \) and \( -4Q \), and (iii) to the right of \( -4Q \).

Case (ii): Between \( +Q \) and \( -4Q \) 
Assume the test charge \( q \) is placed at a point between the two charges. Since both charges will exert attractive or repulsive forces in the same direction (depending on the sign of \( q \)), the forces will not cancel. 
So, equilibrium cannot occur between the charges.

Case (i): To the left of \( +Q \) 
Let the third charge be placed at a distance \( x \) cm to the left of \( +Q \). Then its distance from \( -4Q \) will be \( x + 60 \) cm.
Using Coulomb's law, equating magnitudes of forces for net force to be zero: \[ \frac{Q}{x^2} = \frac{4Q}{(x + 60)^2} \Rightarrow \frac{1}{x^2} = \frac{4}{(x + 60)^2} \Rightarrow \frac{(x + 60)^2}{x^2} = 4 \Rightarrow \frac{x + 60}{x} = 2 \Rightarrow x + 60 = 2x \Rightarrow x = 60 \text{ cm} \]

So, the third charge should be placed 60 cm to the left of \( +Q \). This matches Option (C) since the point is 60 cm from \( +Q \). Let’s confirm directions:
- Force due to \( +Q \) is repulsive.
- Force due to \( -4Q \) is attractive.
Both act in opposite directions if the test charge is to the left of \( +Q \), and they can cancel only in this region.