A wire carrying a 28.0 A current bends through a right angle. Consider 2.00 mm
segments of wire, each 3.00 cm from the bend (see fig). Find magnitude and
direction of the magnetic field these two segments produce at point P, which is
midway between them.
To find the magnitude and direction of the magnetic field produced by the two wire segments at point P, you can use Ampere's law and apply the Biot-Savart law for each segment of the wire.
First, let's calculate the magnetic field generated by each wire segment individually using the Biot-Savart law:
1. Calculate the magnetic field produced by the first wire segment at point P:
- Determine the perpendicular distance (r₁) between the first wire segment and point P. In this case, r₁ = 3.00 cm = 0.03 m.
- The formula for calculating the magnetic field (B) produced by a short straight wire segment is given by:
B₁ = (μ₀ / 4π) * (I₁ * dl₁ × r₁) / r₁³
where μ₀ is the permeability of free space, I₁ is the current in the first wire segment, dl₁ is the length of the first wire segment, and r₁ is the distance between the wire segment and point P.
- Plug in the values to calculate B₁.
2. Calculate the magnetic field produced by the second wire segment at point P:
- Determine the perpendicular distance (r₂) between the second wire segment and point P. Since point P is midway between the two wire segments, r₂ is also 0.03 m.
- Use the same formula as before, but now plug in the values for the second wire segment (current, length, and r₂) to calculate B₂.
Now, to determine the net magnetic field at point P, use Ampere's law, which states that the integral of the magnetic field along a closed loop is equal to the product of the current enclosed by the loop and the permeability of free space (μ₀).
- Construct a circular loop passing through point P, and choose its radius to be greater than the maximum distance from point P to either wire segment (in this case, r > 0.03 m).
- Along the loop, the magnetic field due to each wire segment is directed tangentially in opposite directions (around the loop).
- Since the magnetic field due to each wire segment is at the same distance (r) from point P, they cancel each other out perfectly along any circular loop passing through point P.
- Therefore, the net magnetic field at point P is zero.
In summary, the magnitude of the magnetic field produced by each wire segment at point P can be calculated using the Biot-Savart law, but their directions cancel each other out, resulting in a net magnetic field of zero at point P.