The magnetic force `vecF` on a current carrying conductor of length I in an external magnetic field `vecB` is given by

Options

A. `(IxxvecB)/(vecl)`
B. `(veclxxvecB)/(I)`
C. `I(vecl xx vecB)`
D. `l^(2)vecl xxvecB`

Answer

To derive the expression for the magnetic force \(\vec{F}\) on a current-carrying conductor of length \(L\) in an external magnetic field \(\vec{B}\), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Concept**:
The magnetic force on a charged particle moving in a magnetic field is given by the Lorentz force law. For a single electron, the force can be expressed as:
\[ \vec{F} = -e \vec{v} \times \vec{B} \]
where \(e\) is the charge of the electron, \(\vec{v}\) is the velocity of the electron, and \(\vec{B}\) is the magnetic field.

**Hint**: Recall the Lorentz force law and its application to charged particles.

2. **Considering a Conductor**:
In a conductor carrying current \(I\), there are many charge carriers (like electrons). The total force on the conductor can be derived from the force on individual charge carriers.

3. **Calculating the Total Force**:
Let \(N\) be the number of charge carriers per unit volume, \(A\) be the cross-sectional area of the conductor, and \(L\) be its length. The total number of charge carriers in the conductor is \(N \cdot A \cdot L\).

4. **Relating Current to Charge Movement**:
The current \(I\) is related to the charge movement as follows:
\[ I = N \cdot A \cdot e \cdot v_d \]
where \(v_d\) is the drift velocity of the charge carriers.

5. **Expressing the Force on the Conductor**:
The total magnetic force on the conductor can be expressed as:
\[ \vec{F} = N \cdot A \cdot e \cdot \vec{v_d} \times \vec{B} \]

6. **Substituting for Current**:
From the previous equation, we can substitute \(N \cdot A \cdot e \cdot v_d\) with \(I\):
\[ \vec{F} = I \cdot \vec{L} \times \vec{B} \]
where \(\vec{L}\) is a vector of length \(L\) in the direction of the current.

7. **Final Expression**:
Thus, the magnetic force on a current-carrying conductor in a magnetic field is given by:
\[ \vec{F} = I \cdot \vec{L} \times \vec{B} \]

### Conclusion:
The expression for the magnetic force \(\vec{F}\) on a current-carrying conductor of length \(L\) in an external magnetic field \(\vec{B}\) is:
\[ \vec{F} = I \cdot \vec{L} \times \vec{B} \]