Two parallel wires carry currents in the same direction. I1=3A and I2=1A and the distance between the conductors is 4 cm. The magnetic field midway between these two wires is
Subtract the magnetic fields (since they are in opposite directions) due to the separate wires at a distance of 2 cm from each wire.
To find the magnetic field midway between two parallel wires, you can use Ampere's Law.
Ampere's Law states that the magnetic field around a closed loop is proportional to the current passing through the loop. In mathematical terms, it can be expressed as:
∮B · dl = μ0 * I,
where B is the magnetic field, dl is an infinitesimal length along the closed loop, μ0 is the permeability of free space (a constant), and I is the current passing through the loop.
To apply Ampere's Law to this situation, you can consider a circular loop centered between the two wires. The circular loop will have a radius equal to the distance between the conductors divided by 2.
Now let's calculate the magnetic field at the midpoint between the two wires:
1. Determine the current passing through the circular loop:
Since both wires carry currents in the same direction, you can simply add the currents together:
I = I1 + I2 = 3A + 1A = 4A.
2. Calculate the circumference of the circular loop:
The circumference of a circle is given by 2πr, where r is the radius.
In this case, the radius is half the distance between the conductors, which is 4 cm/2 = 2 cm = 0.02 m.
So, the circumference is 2π(0.02 m) = 0.04π m.
3. Apply Ampere's Law:
∮B · dl = μ0 * I,
B * 0.04π m = (4π × 10^-7 T·m/A) * 4A.
4. Solve for B:
B = (4π × 10^-7 T·m/A) * 4A / (0.04π m)
= 4π × 10^-7 T·m/A / 0.04 m
= 4 × 10^-7 T.
Therefore, the magnetic field midway between the two wires is 4 × 10^-7 Tesla (T).