Question

In a vertically upwards magnetic field \(\vec{B}\), a positively charged particle is projected horizontally eastwards. What will be the direction of force on the particle?

Hint:

For positive charges, use Fleming's Left-Hand Rule directly. If the charge were negative (like an electron), the force direction would be opposite to what the rule indicates.

Answer 1

Step 1: Understanding the Concept:
The force experienced by a charged particle 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.
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:
\[ \vec{F} = q(\vec{v} \times \vec{B}) \] The direction of the force is given by the direction of the cross product \(\vec{v} \times \vec{B}\). We can use the right-hand rule for the cross product (since the charge is positive).
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 (\(+\hat{j}\)).
- Vertically upwards is along the positive z-axis (\(+\hat{k}\)).
From the problem statement:
- The velocity vector \(\vec{v}\) is horizontally eastwards: \(\vec{v} = v\hat{i}\).
- The magnetic field vector \(\vec{B}\) is vertically upwards: \(\vec{B} = B\hat{k}\).
Now, we calculate the cross product:
\[ \vec{v} \times \vec{B} = (v\hat{i}) \times (B\hat{k}) = vB (\hat{i} \times \hat{k}) \] Using the cyclic property of unit vectors for cross product (\(\hat{i} \times \hat{j} = \hat{k}\), \(\hat{j} \times \hat{k} = \hat{i}\), \(\hat{k} \times \hat{i} = \hat{j}\)), we know that \(\hat{i} \times \hat{k} = -\hat{j}\).
\[ \vec{v} \times \vec{B} = vB (-\hat{j}) \] The direction \(-\hat{j}\) corresponds to the South.
Alternative Method (Fleming's Left-Hand Rule):
- Point your Forefinger in the direction of the Magnetic Field (Upwards).
- Point your Middle finger in the direction of the Current (or motion of positive charge, which is Eastwards).
- Your Thumb will point in the direction of the Force, which is towards the South.
Step 4: Final Answer:
The direction of the force on the positively charged particle is towards the South.