JEE Physics 2024 Question with Solution

Given: A particle of charge $-q$ and mass $m$ moves in a circular orbit of radius $r$ around an infinitely long line charge of linear charge density $+\lambda$.

Find: The time period $T$.

For an infinitely long line charge, the electric field at distance $r$ is

$$E = \frac{\lambda}{2\pi \epsilon_0 r} = \frac{2k\lambda}{r}$$

The magnitude of the attractive electrostatic force on the particle is

$$F_e = qE = q\left(\frac{2k\lambda}{r}\right) = \frac{2kq\lambda}{r}$$

This provides the centripetal force, so

$$\frac{mv^2}{r} = \frac{2kq\lambda}{r}$$

Hence,

$$mv^2 = 2kq\lambda$$

so

$$v = \sqrt{\frac{2kq\lambda}{m}}$$

Now use

$$T = \frac{2\pi r}{v}$$

Substituting for $v$,

$$T = \frac{2\pi r}{\sqrt{\frac{2kq\lambda}{m}}} = 2\pi r\sqrt{\frac{m}{2kq\lambda}}$$

Therefore, the time period is $T = 2\pi r\sqrt{\frac{m}{2kq\lambda}}$.

The solution states that this matches the second option. However, the listed options omit $\lambda$. Using the source solution's own option mapping, the defensible answer is B.

Given: Circular motion under electrostatic attraction due to an infinite line charge.

Find: The correct expression for $T$.

1. Electric field due to an infinite line charge:

$$E = \frac{\lambda}{2\pi \epsilon_0 r}$$

Using $k = \frac{1}{4\pi\epsilon_0}$,

$$E = \frac{2k\lambda}{r}$$

2. Force on charge magnitude $q$:

$$F = qE = \frac{2kq\lambda}{r}$$

Since the charge is negative, the force is toward the line, which is exactly the required centripetal direction.

3. Equate with centripetal force:

$$\frac{mv^2}{r} = \frac{2kq\lambda}{r}$$

Cancel $r$:

$$mv^2 = 2kq\lambda$$

Thus,

$$v^2 = \frac{2kq\lambda}{m}$$

and

$$v = \sqrt{\frac{2kq\lambda}{m}}$$

4. Time period of circular motion:

$$T = \frac{2\pi r}{v}$$

Therefore,

$$T = 2\pi r\sqrt{\frac{m}{2kq\lambda}}$$

This is the physically correct expression extracted from the solution. Because the options as provided do not include $\lambda$, there is a discrepancy in the source. The solution still identifies the second option, so the answer is taken as B.