Two mutually perpendicular conductors carrying currents `I_(1)` and `I_(2)` lie in one plane. Locus of the point at which the magnetic induction is zero, is a:

Options

A. circle with centre as the point of intersection of the conductor
B. parabola with vertex as the point of intersection of the conductors
C. straight line passing through the point of intersection of the conductors
D. rectangular hyperbola

Explanation

To solve the problem, we need to determine the locus of points where the magnetic induction (magnetic field) is zero due to two mutually perpendicular conductors carrying currents \(I_1\) and \(I_2\).

### Step-by-Step Solution:

1. **Understanding the Magnetic Field due to a Long Straight Conductor**:
The magnetic field \(B\) at a distance \(r\) from a long straight conductor carrying current \(I\) is given by the formula:
\[ B = \frac{\mu_0 I}{2 \pi r} \]
where \(\mu_0\) is the permeability of free space.

2. **Setting Up the Coordinate System**:
Assume that the first conductor carrying current \(I_1\) is along the y-axis and the second conductor carrying current \(I_2\) is along the x-axis. Let the point \(P\) where we want to find the magnetic field be at coordinates \((x, y)\).

3. **Calculating the Magnetic Field at Point P**:
- The magnetic field \(B_1\) due to the conductor along the y-axis at point \(P\) (distance \(y\) from the first conductor) is directed into the plane (using the right-hand rule):
\[ B_1 = \frac{\mu_0 I_1}{2 \pi y} \hat{k} \]
- The magnetic field \(B_2\) due to the conductor along the x-axis at point \(P\) (distance \(x\) from the second conductor) is directed out of the plane:
\[ B_2 = -\frac{\mu_0 I_2}{2 \pi x} \hat{k} \]

4. **Setting the Net Magnetic Field to Zero**:
For the net magnetic field \(B\) to be zero, we set the sum of the two magnetic fields equal to zero:
\[ B_1 + B_2 = 0 \]
Substituting the expressions for \(B_1\) and \(B_2\):
\[ \frac{\mu_0 I_1}{2 \pi y} - \frac{\mu_0 I_2}{2 \pi x} = 0 \]

5. **Simplifying the Equation**:
We can cancel \(\frac{\mu_0}{2 \pi}\) from both sides:
\[ \frac{I_1}{y} = \frac{I_2}{x} \]
Rearranging gives:
\[ y = \frac{I_1}{I_2} x \]

6. **Identifying the Locus**:
The equation \(y = \frac{I_1}{I_2} x\) represents a straight line passing through the origin (0,0) with a slope of \(\frac{I_1}{I_2}\). This line is the locus of points where the magnetic induction is zero.

### Conclusion:
The locus of the point at which the magnetic induction is zero is a straight line passing through the point of intersection of the conductors.