Question 105: Electrostats & Capacitance

Question

PQRS is square of side $$1 \mathrm{~m}$$. A charge of $$100 \mu \mathrm{C}$$ is placed at the centre of the square. Then the work done to take $$3 \mu \mathrm{C}$$ charge from the corner $$\mathrm{P}$$ to the corner $$\mathrm{R}$$.

Accepted Answer

To find the work done in moving a charge from one point to another in an electric field, we need to first understand that work done is given by the difference in potential energy between the two points, which can also be interpreted as the potential difference times the charge being moved, i.e.,
$ Work = q \cdot \Delta V $
where $ q $ is the charge being moved, and $ \Delta V $ is the potential difference between the initial and final points.
For a charge $ Q $ placed at a point, the potential $ V $ at a distance $ r $ from the charge is given by Coulomb's law as:
$ V = \frac{kQ}{r} $
where $ k $ is Coulomb's constant ($ k = 9 	imes 10^{9} \, \mathrm{Nm^2/C^2} $).
In the case of the square PQRS with a charge of $ 100 \mu C $ at the center and sides of $1 \mathrm{~m}$, the potential at any corner due to the central charge would be the same because all corners are equidistant from the center.
Since $ PQRS $ is a square of side $1 \mathrm{~m}$, the distance from the center to any corner (the diagonal of the square, which is half the diagonal of the square) is:
$ d = \frac{\sqrt{2}}{2} $
The potential at any corner (or the work per unit charge to move a charge from infinity to that point) due to the central charge $ Q = 100 \mu C = 100 	imes 10^{-6} C $ is:
$ V = \frac{kQ}{d} = \frac{9 	imes 10^{9} \cdot 100 	imes 10^{-6}}{\sqrt{2}/2} = 9 \sqrt{2} 	imes 10^{5} \, V $
Since the potential everywhere along the path (any path on the square plane) from P to R due to the central charge is constant because the distance from each point on the path to the charge is constant, the potential difference $ \Delta V $ between corners P and R is zero. Therefore, the work done in moving any charge from P to R is:
$ Work = q \cdot \Delta V = q \cdot 0 = 0 $
The correct answer is therefore:
Option C: Zero.

Answer

$$9 \sqrt{2} 	imes 10^5 \mathrm{~J}$$

Answer

$$4.5 	imes 10^5 \mathrm{~J}$$

Answer

Zero

Answer

$$1.8 	imes 10^5 \mathrm{~J}$$

Electrostats & Capacitance

PQRS is square of side $1 \mathrm{~m}. A charge of 100 \mu \mathrm{C} is placed at the centre of the square. Then the work done to take 3 \mu \mathrm{C} charge from the corner \mathrm{P} to the corner \mathrm{R}$.

Choose the correct answer: