Question

An electron moves through a uniform magnetic field \( \vec{B} = B_0 \hat{i} + 2B_0 \hat{j} \, \text{T} \). At a particular instant of time, the velocity of the electron is \( \vec{u} = 3\hat{i} + 5\hat{j} \, \text{m/s} \). If the magnetic force acting on the electron is \( \vec{F} = 5e k \, \text{N} \), where \( e \) is the charge of the electron, then the value of \( B_0 \) is _____ T

Answer 1

Correct Answer: 5

The magnetic force on a charged particle in a magnetic field is described by \(|\vec{F}| = q|\vec{u} \times \vec{B}|\).

Given: \( \vec{F} = 5e \hat{k} \, \text{N} \), \( q = -e \) (electron charge), \( \vec{u} = 3\hat{i} + 5\hat{j} \, \text{m/s} \), and \( \vec{B} = B_0 \hat{i} + 2B_0 \hat{j} \, \text{T} \).

The cross product \(\vec{u} \times \vec{B}\) is calculated as:

\[|\vec{u} \times \vec{B}| = \left|\begin{array}{ccc}\hat{i} & \hat{j} & \hat{k} \\ 3 & 5 & 0 \\ B_0 & 2B_0 & 0 \end{array}\right|\]

Evaluating the determinant yields \(\hat{0}\hat{i} - \hat{0}\hat{j} + (3 \cdot 2B_0 - 5 \cdot B_0) \hat{k}\):

\[|\vec{u} \times \vec{B}| = (6B_0 - 5B_0)\hat{k} = B_0\hat{k}\]

Therefore, \(|\vec{F}| = e|\vec{u} \times \vec{B}| = eB_0\).

Since \(|\vec{F}| = 5e\), we have:

\[eB_0 = 5e\]

Solving for \(B_0\):

\[B_0 = 5\]

The calculated value \(B_0 = 5 \, \text{T}\) is consistent with the given range \([5, 5]\).