Question

Two long straight parallel wires separated by 20 cm carry 5 A and 10 A current respectively, in the same direction. Find the magnitude and direction of the net magnetic field at a point midway between them.

Hint:

When calculating the magnetic field due to parallel currents, use Ampere's law and apply the right-hand rule to determine the direction of the field. The fields due to currents in the same direction will oppose each other.

Answer 1

Step 1: The magnetic field \( B \) at a point due to a long straight current-carrying conductor is given by Ampere's law:
\[ B = \frac{\mu_0 I}{2 \pi r} \] where:
\( B \) is the magnetic field at a distance \( r \) from the wire,
\( I \) is the current in the wire,
\( r \) is the distance from the wire,
\( \mu_0 \) is the permeability of free space (\( \mu_0 = 4\pi \times 10^{-7} \, \text{}^2 \)).
Step 2: For two wires carrying currents in the same direction, the magnetic field due to each wire at a point midway between them (which is at a distance \( r = 10 \, \text{cm} = 0.1 \, \text{m} \) from each wire) can be calculated using the formula above.
For the first wire with current \( I_1 = 5 \, \text{A} \):
\[ B_1 = \frac{\mu_0 I_1}{2 \pi r} = \frac{(4\pi \times 10^{-7}) \times 5}{2 \pi \times 0.1} = 1 \times 10^{-5} \, \text{T} \] For the second wire with current \( I_2 = 10 \, \text{A} \):
\[ B_2 = \frac{\mu_0 I_2}{2 \pi r} = \frac{(4\pi \times 10^{-7}) \times 10}{2 \pi \times 0.1} = 2 \times 10^{-5} \, \text{T} \] Step 3: Since the currents are in the same direction, the magnetic fields due to the two wires at the midway point will be in opposite directions. Therefore, the net magnetic field at the point midway between the wires is the difference between the two fields:
\[ B_{\text{net}} = B_2 - B_1 = 2 \times 10^{-5} - 1 \times 10^{-5} = 1 \times 10^{-5} \, \text{T} \] Step 4: The direction of the magnetic field can be determined by using the right-hand rule. The magnetic field due to the first wire is directed into the page, and the magnetic field due to the second wire is directed out of the page. Since the fields are in opposite directions, the net magnetic field will be directed out of the page.
Thus, the magnitude of the net magnetic field is \( 1 \times 10^{-5} \, \text{T} \), and the direction is out of the page.