A thin straight horizontal wire of length `0.2 m` whose mass is `10^(-4) kg` floats in a magnetic induction field when a current of `10 ampere` is passed throught it.To make this possible, what should be minimum magnetic strength?

Answer

To solve the problem, we need to determine the minimum magnetic field strength required for a thin straight horizontal wire to float when a current is passed through it.

### Step-by-Step Solution:

1. **Identify the Forces Acting on the Wire**:
- The wire has a weight acting downwards due to gravity, which can be calculated using the formula:
\[ \text{Weight} (W) = m \cdot g \]
- The magnetic force acting upwards due to the magnetic field when current flows through the wire is given by:
\[ F_m = B \cdot I \cdot L \]
where \( B \) is the magnetic field strength, \( I \) is the current, and \( L \) is the length of the wire.

2. **Set Up the Equation for Floating Condition**:
- For the wire to float, the magnetic force must equal the weight of the wire:
\[ F_m = W \]
- Therefore, we can write:
\[ B \cdot I \cdot L = m \cdot g \]

3. **Substitute the Given Values**:
- Given:
- Length of the wire, \( L = 0.2 \, \text{m} \)
- Mass of the wire, \( m = 10^{-4} \, \text{kg} \)
- Current, \( I = 10 \, \text{A} \)
- Acceleration due to gravity, \( g = 10 \, \text{m/s}^2 \)
- Substituting these values into the equation:
\[ B \cdot 10 \cdot 0.2 = 10^{-4} \cdot 10 \]

4. **Simplify the Equation**:
- This simplifies to:
\[ B \cdot 2 = 10^{-3} \]
- Rearranging gives:
\[ B = \frac{10^{-3}}{2} \]

5. **Calculate the Magnetic Field Strength**:
- Performing the division:
\[ B = 0.5 \times 10^{-3} \, \text{T} = 0.5 \, \text{mT} \]

6. **Final Answer**:
- The minimum magnetic field strength required for the wire to float is:
\[ B = 0.5 \, \text{milliTesla} \]

### Summary:
The minimum magnetic strength required for the wire to float is **0.5 milliTesla**.

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