Question

An electron projected towards East is deflected towards North by a magnetic field. The direction of magnetic field may be

• A. towards West
• B. towards South
• C. perpendicular to the plane upwards
• D. perpendicular to the plane downwards

Hint:

For questions involving forces on charged particles, always check the sign of the charge. For electrons (negative charge), the direction of the magnetic force is opposite to what the standard right-hand rule would predict for a positive charge. Using Fleming's Left-Hand Rule, point the current finger opposite to the electron's velocity.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
The force experienced by a charge moving in a magnetic field is given by the Lorentz force. The direction of this force can be determined using Fleming's Left-Hand Rule or the vector cross product. It's crucial to remember that the electron has a negative charge, which reverses the direction of the force compared to a positive charge.

Step 2: Key Formula or Approach:
The magnetic force \(\vec{F}\) on a charge \(q\) moving with velocity \(\vec{v}\) in a magnetic field \(\vec{B}\) is given by: \[ \vec{F} = q(\vec{v} \times \vec{B}) \] For an electron, \(q = -e\). So, \(\vec{F} = -e(\vec{v} \times \vec{B})\).
This means the direction of the force on an electron is opposite to the direction given by the right-hand rule for \(\vec{v} \times \vec{B}\). Alternatively, we can use Fleming's Left-Hand Rule, but remember to point the current direction (middle finger) opposite to the electron's velocity.

Step 3: Detailed Explanation:
Let's define the directions using a standard coordinate system:
- East is along the positive x-axis (\(+\hat{i}\)).
- North is along the positive y-axis (\(+\hat{j}\)).
- Upwards is along the positive z-axis (\(+\hat{k}\)).
- Downwards is along the negative z-axis (\(-\hat{k}\)).
Given information:
- Direction of electron's velocity, \(\vec{v}\), is towards East (\(+\hat{i}\)).
- Direction of magnetic force, \(\vec{F}\), is towards North (\(+\hat{j}\)).
Using the formula for the force on an electron: \[ \vec{F} = -e(\vec{v} \times \vec{B}) \] The direction of \(\vec{F}\) is North (\(+\hat{j}\)). This implies that the direction of \((\vec{v} \times \vec{B})\) must be opposite, i.e., towards South (\(-\hat{j}\)).
So we have: \[ (\text{direction of } \vec{v}) \times (\text{direction of } \vec{B}) = \text{South} \] \[ (+\hat{i}) \times (\text{direction of } \vec{B}) = -\hat{j} \] From the properties of vector cross products, we know that \(\hat{i} \times (-\hat{k}) = -\hat{j}\).
Therefore, the direction of the magnetic field \(\vec{B}\) must be along \(-\hat{k}\), which is perpendicular to the plane (East-North plane) and directed downwards.

Step 4: Final Answer:
The direction of the magnetic field is perpendicular to the plane downwards.