Question

Inside a solenoid of radius \( 0.5 \) m, the magnetic field is changing at a rate of \( 50 \times 10^{-6} \) T/s. The acceleration of an electron placed at a distance of \( 0.3 \) m from the axis of the solenoid will be:

• A. \( 23 \times 10^6 \) m/s\(^2\)
• B. \( 26 \times 10^6 \) m/s\(^2\)
• C. \( 1.3 \times 10^9 \) m/s\(^2\)
• D. \( 26 \times 10^9 \) m/s\(^2\)

Hint:

The changing magnetic field inside a solenoid induces an electric field, which exerts a force on charged particles.

Answer 1

The Correct Option is A

Step 1: {Using Faraday's Law}
\[ {Induced emf} = \frac{-d\Phi}{dt} = B \cdot A \] Since \( A = \pi r^2 \), we get: \[ \varepsilon = -\pi r^2 \frac{dB}{dt} \] Step 2: {Finding the Electric Field}
\[ E = \frac{\varepsilon}{d} = \frac{\pi r^2}{d} \frac{dB}{dt} \] Step 3: {Finding Acceleration}
\[ a = \frac{eE}{m} \] Substituting values, \[ a = \frac{1.6 \times 10^{-19}}{9.1 \times 10^{-31}} \times \frac{\pi (0.5)^2}{0.3} \times 50 \times 10^{-6} \] \[ = 23 \times 10^6 { m/s}^2 \] Thus, the correct answer is \( 23 \times 10^6 \) m/s\(^2\).