Question

A magnetic field of (10^-4\(\hat{k}\))T exerts a force of (4\(\hat{i}\)-3\(\hat{j}\))x10^-12N on a particle having a charge of 10^-9 C. The speed of the particle is:

• A. 40 m/s
• B. 40\(\sqrt{2}\) m/s
• C. 50 m/s
• D. 50\(\sqrt{3}\) m/s
• E. 100\(\sqrt{2}\)m/s

Answer 1

The Correct Option is C

Given parameters:

• Magnetic field \( \vec{B} = 10^{-4} \hat{k} \, \text{T} \)
• Force \( \vec{F} = (4\hat{i} - 3\hat{j}) \times 10^{-12} \, \text{N} \)
• Charge \( q = 10^{-9} \, \text{C} \)

 

Lorentz force equation: \[ \vec{F} = q(\vec{v} \times \vec{B}) \]

Velocity components: \[ (4\hat{i} - 3\hat{j}) \times 10^{-12} = 10^{-9}(\vec{v} \times 10^{-4}\hat{k}) \] \[ \vec{v} \times \hat{k} = (4\hat{i} - 3\hat{j}) \times 10 \]

Solve for velocity: \[ \vec{v} = 40\hat{j} + 30\hat{i} \] \[ |\vec{v}| = \sqrt{30^2 + 40^2} = 50 \, \text{m/s} \]

Thus, the correct option is (C): 50 m/s.

Answer 2

1. Recall the magnetic force equation:

The magnetic force (F) on a charged particle moving in a magnetic field (B) is given by:

\[\vec{F} = q(\vec{v} \times \vec{B})\]

where:

• q is the charge of the particle
• v is the velocity of the particle
• B is the magnetic field

2. Define the given information:

\[\vec{B} = 10^{-4}\hat{k} \, T\]

\[\vec{F} = (4\hat{i} - 3\hat{j}) \times 10^{-12} \, N\]

\[q = 10^{-9} \, C\]

3. Set up the equation and solve for the velocity:

Let \(\vec{v} = v_x\hat{i} + v_y\hat{j} + v_z\hat{k}\). Substituting into the magnetic force equation:

\[(4\hat{i} - 3\hat{j}) \times 10^{-12} = 10^{-9}((v_x\hat{i} + v_y\hat{j} + v_z\hat{k}) \times 10^{-4}\hat{k})\]

Calculate the cross product:

\[(4\hat{i} - 3\hat{j}) \times 10^{-12} = 10^{-9} \times 10^{-4}(v_x(\hat{i} \times \hat{k}) + v_y(\hat{j} \times \hat{k}) + v_z(\hat{k} \times \hat{k}))\]

\[(4\hat{i} - 3\hat{j}) \times 10^{-12} = 10^{-13}(-v_x\hat{j} + v_y\hat{i})\]

Equate the components:

\[4 \times 10^{-12} = 10^{-13}v_y \implies v_y = 40 \, m/s\]

\[-3 \times 10^{-12} = -10^{-13}v_x \implies v_x = 30 \, m/s\]

The velocity vector is \(\vec{v} = 30\hat{i} + 40\hat{j}\) (the z-component is indeterminate as \(v_z\) disappears in the cross product).

4. Calculate the speed:

The speed is the magnitude of the velocity vector:

\[|\vec{v}| = \sqrt{v_x^2 + v_y^2} = \sqrt{30^2 + 40^2} = \sqrt{900 + 1600} = \sqrt{2500} = 50 \, m/s\]