A particle of mass `0.6g` and having charge of `25 n_(C)` is moving horizontally with a uniform velocity `1.2xx10^(4)ms^(-1)` in a uniform magnetic field, then the value of the magnetic induction is `(g=10ms^(-2))`

Options

A. Zero
B. 10 T
C. 20 T
D. 200 T

Answer

To solve the problem, we need to find the magnetic induction (magnetic field strength) \( B \) for a charged particle moving in a magnetic field. The magnetic force acting on the particle is equal to the weight of the particle.

### Step-by-Step Solution:

1. **Convert the mass and charge to standard units:**
- Mass \( m = 0.6 \, \text{g} = 0.6 \times 10^{-3} \, \text{kg} \)
- Charge \( Q = 25 \, \text{nC} = 25 \times 10^{-9} \, \text{C} \)

2. **Identify the given values:**
- Velocity \( V = 1.2 \times 10^{4} \, \text{m/s} \)
- Acceleration due to gravity \( g = 10 \, \text{m/s}^2 \)

3. **Write the equation for the magnetic force:**
The magnetic force \( F_B \) acting on the charged particle is given by:
\[ F_B = QVB \]
where \( B \) is the magnetic induction.

4. **Write the equation for the weight of the particle:**
The weight \( W \) of the particle is given by:
\[ W = mg \]

5. **Set the magnetic force equal to the weight:**
Since the magnetic force balances the weight of the particle:
\[ QVB = mg \]

6. **Rearrange the equation to solve for \( B \):**
\[ B = \frac{mg}{QV} \]

7. **Substitute the known values into the equation:**
\[ B = \frac{(0.6 \times 10^{-3} \, \text{kg})(10 \, \text{m/s}^2)}{(25 \times 10^{-9} \, \text{C})(1.2 \times 10^{4} \, \text{m/s})} \]

8. **Calculate the numerator:**
\[ \text{Numerator} = 0.6 \times 10^{-3} \times 10 = 6 \times 10^{-3} \, \text{kg m/s}^2 \]

9. **Calculate the denominator:**
\[ \text{Denominator} = (25 \times 10^{-9})(1.2 \times 10^{4}) = 30 \times 10^{-5} = 3 \times 10^{-4} \, \text{C m/s} \]

10. **Now substitute back into the equation for \( B \):**
\[ B = \frac{6 \times 10^{-3}}{3 \times 10^{-4}} = 20 \, \text{T} \]

### Final Answer:
The value of the magnetic induction \( B \) is \( 20 \, \text{T} \).

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