A proton is moving with a velocity of `5 xx 10^(5) m//s` along the Y-direction. It is acted upon by an electric field of intensity `10^(5)` V/m along the X-direction and a magnetic field of `1 Wb//m^(2)` along the Z-direction. Then the Lorentz force on the particle is :

Options

A. zero
B. `8.0 xx 10^(-4)` N along X-axis
C. `9.6 xx 10^(-14)` along Z-axis
D. `9.6 xx 10^(-14)` along X-axis

Answer

To find the Lorentz force acting on a proton moving in an electric and magnetic field, we can follow these steps:

### Step 1: Identify the Given Values
- Velocity of the proton, \( v = 5 \times 10^5 \, \text{m/s} \) (along the Y-direction)
- Electric field intensity, \( E = 10^5 \, \text{V/m} \) (along the X-direction)
- Magnetic field intensity, \( B = 1 \, \text{Wb/m}^2 \) (along the Z-direction)
- Charge of the proton, \( Q = 1.6 \times 10^{-19} \, \text{C} \)

### Step 2: Calculate the Electric Force
The electric force \( F_E \) acting on the proton can be calculated using the formula:
\[ F_E = Q \cdot E \]
Substituting the values:
\[ F_E = (1.6 \times 10^{-19} \, \text{C}) \cdot (10^5 \, \text{V/m}) = 1.6 \times 10^{-14} \, \text{N} \]
The direction of the electric force is along the X-direction (positive X-direction).

### Step 3: Calculate the Magnetic Force
The magnetic force \( F_M \) can be calculated using the formula:
\[ F_M = Q \cdot (v \times B) \]
First, we need to determine the direction of the cross product \( v \times B \):
- The velocity \( v \) is in the Y-direction (denote as \( \hat{j} \)).
- The magnetic field \( B \) is in the Z-direction (denote as \( \hat{k} \)).

Using the right-hand rule:
\[ \hat{j} \times \hat{k} = \hat{i} \]
Thus, the direction of \( v \times B \) is in the X-direction.

Now, we can calculate the magnitude of the magnetic force:
\[ F_M = Q \cdot v \cdot B \]
Substituting the values:
\[ F_M = (1.6 \times 10^{-19} \, \text{C}) \cdot (5 \times 10^5 \, \text{m/s}) \cdot (1 \, \text{Wb/m}^2) \]
Calculating this gives:
\[ F_M = 8 \times 10^{-14} \, \text{N} \]
The direction of the magnetic force is also along the X-direction.

### Step 4: Calculate the Total Lorentz Force
The total Lorentz force \( F \) is the sum of the electric and magnetic forces:
\[ F = F_E + F_M \]
Substituting the values:
\[ F = (1.6 \times 10^{-14} \, \text{N}) + (8 \times 10^{-14} \, \text{N}) = 9.6 \times 10^{-14} \, \text{N} \]
The direction of the total Lorentz force is in the X-direction.

### Final Answer
The Lorentz force on the proton is:
\[ F = 9.6 \times 10^{-14} \, \text{N} \, \text{in the X-direction} \]

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