Two long parallel transmission lines `40.0 cm` apart carry `25.0 A` and `75.0 A `currents. Find an locations where the net magnetic field of the two wires is zero if these currents are in
(a) the same direction (b) the opposite direction

Explanation

To solve the problem of finding the locations where the net magnetic field of two long parallel transmission lines is zero, we will analyze two cases: (a) when the currents are in the same direction and (b) when the currents are in opposite directions.

### Given Data:
- Distance between the wires, \( d = 40.0 \, \text{cm} = 0.4 \, \text{m} \)
- Current in wire 1, \( I_1 = 25.0 \, \text{A} \)
- Current in wire 2, \( I_2 = 75.0 \, \text{A} \)

### (a) Currents in the Same Direction

1. **Understanding Magnetic Fields**:
- For two parallel wires carrying current in the same direction, the magnetic field between the wires will be directed downwards, and outside the wires, the magnetic field will be directed upwards.

2. **Setting Up the Equation**:
- Let \( R \) be the distance from wire 1 (25 A) to the point where the magnetic field is zero. Therefore, the distance from wire 2 (75 A) to this point will be \( (0.4 - R) \).

3. **Magnetic Field Formula**:
- The magnetic field due to a long straight current-carrying wire at a distance \( r \) is given by:
\[ B = \frac{\mu_0 I}{2 \pi r} \]
- Where \( \mu_0 \) is the permeability of free space.

4. **Equating the Magnetic Fields**:
- At the point where the magnetic field is zero, the magnetic fields due to both wires must be equal in magnitude:
\[ B_1 = B_2 \]
- Therefore,
\[ \frac{\mu_0 I_1}{2 \pi R} = \frac{\mu_0 I_2}{2 \pi (0.4 - R)} \]
- The \( \mu_0 \) and \( 2 \pi \) cancel out:
\[ \frac{I_1}{R} = \frac{I_2}{0.4 - R} \]

5. **Substituting the Values**:
- Substituting \( I_1 = 25 \, \text{A} \) and \( I_2 = 75 \, \text{A} \):
\[ \frac{25}{R} = \frac{75}{0.4 - R} \]

6. **Cross-Multiplying**:
- Cross-multiplying gives:
\[ 25(0.4 - R) = 75R \]
- Simplifying:
\[ 10 - 25R = 75R \]
\[ 10 = 100R \]
\[ R = 0.1 \, \text{m} = 10 \, \text{cm} \]

7. **Finding the Position**:
- The point where the magnetic field is zero is \( 10 \, \text{cm} \) from the wire carrying \( 25 \, \text{A} \) and \( 30 \, \text{cm} \) from the wire carrying \( 75 \, \text{A} \).

### (b) Currents in Opposite Directions

1. **Understanding Magnetic Fields**:
- For two parallel wires carrying currents in opposite directions, the magnetic field between the wires will be directed upwards, and outside the wires, the magnetic field will be directed downwards.

2. **Setting Up the Equation**:
- Again let \( R_1 \) be the distance from wire 1 (25 A) to the point where the magnetic field is zero. The distance from wire 2 (75 A) will then be \( (0.4 + R_1) \).

3. **Equating the Magnetic Fields**:
- At the point where the magnetic field is zero:
\[ B_1 = B_2 \]
- Therefore,
\[ \frac{\mu_0 I_1}{2 \pi R_1} = \frac{\mu_0 I_2}{2 \pi (0.4 + R_1)} \]

4. **Substituting the Values**:
- Substituting \( I_1 = 25 \, \text{A} \) and \( I_2 = 75 \, \text{A} \):
\[ \frac{25}{R_1} = \frac{75}{0.4 + R_1} \]

5. **Cross-Multiplying**:
- Cross-multiplying gives:
\[ 25(0.4 + R_1) = 75R_1 \]
- Simplifying:
\[ 10 + 25R_1 = 75R_1 \]
\[ 10 = 50R_1 \]
\[ R_1 = 0.2 \, \text{m} = 20 \, \text{cm} \]

6. **Finding the Position**:
- The point where the magnetic field is zero is \( 20 \, \text{cm} \) from the wire carrying \( 25 \, \text{A} \) and \( 20 \, \text{cm} \) from the wire carrying \( 75 \, \text{A} \).

### Summary of Results:
- (a) For currents in the same direction, the point where the magnetic field is zero is \( 10 \, \text{cm} \) from the \( 25 \, \text{A} \) wire.
- (b) For currents in opposite directions, the point where the magnetic field is zero is \( 20 \, \text{cm} \) from the \( 25 \, \text{A} \) wire.