a) The Earth magnetic field on the equator is directed horizontally from South to North; its strength is 5∗10−5T
. A power line with the direct current (i.e. always in the same direction) of 100 A runs also horizontally from East to West. You measure the total magnetic field at 20 m directly below the power line. What will be the total magnetic field strength and direction? µ0=4π∗10−7T
m/A.
b) Two parallel wires with currents I1=1A and I2=2A running in the same direction are separated by a distance of 10 cm. What is the magnitude of the force with which the wire with the current I1 acts on 10-m long segment of the wire with the current I2?
a) To find the total magnetic field strength at a point below the power line, we can use the formula for the magnetic field produced by a current-carrying wire:
B = (µ0 * I) / (2π * r)
where B is the magnetic field strength, µ0 is the permeability of free space, I is the current in the wire, and r is the distance from the wire.
For the Earth's magnetic field, we know the strength is 5∗10^(-5) T and directed horizontally from South to North.
For the power line, the current is 100 A and running horizontally from East to West.
Plugging in the values, we get:
B_earth = 5∗10^(-5) T
B_line = (4π∗10^(-7) T m/A * 100 A) / (2π * 20 m) = 10^(-4) T
To find the total magnetic field strength and direction at the point below the power line, we need to sum the vectors:
B_total = sqrt((B_earth)^2 + (B_line)^2) = sqrt((5∗10^(-5))^2 + (10^(-4))^2) = sqrt(25∗10^(-10) + 100∗10^(-8)) = sqrt(25∗10^(-10) + 1∗10^(-6)) = sqrt(26∗10^(-10)) = sqrt(26)∗10^(-5) = 5.1∗10^(-5) T
The direction of the total magnetic field can be found by finding the angle θ:
tan(θ) = B_earth / B_line
θ = arctan(B_earth / B_line) = arctan(5∗10^(-5) / 10^(-4)) = arctan(0.5) ≈ 26.57 degrees
Therefore, the total magnetic field strength at the point below the power line is 5.1∗10^(-5) T directed at an angle of approximately 26.57 degrees from the horizontal.
b) To find the force with which the wire with current I1 acts on the wire with current I2, we can use the formula for the force between two parallel wires:
F = (µ0 * I1 * I2 * L) / (2π * d)
where F is the force, µ0 is the permeability of free space, I1 and I2 are the currents in the wires, L is the length of the wire, and d is the distance between the wires.
Plugging in the values, we get:
F = (4π∗10^(-7) T m/A * 1 A * 2 A * 10 m) / (2π * 0.1 m) = (8∗10^(-7) T m * 10 A) / 0.2 m = 0.4∗10^(-5) N = 4∗10^(-6) N
Therefore, the magnitude of the force with which the wire with current I1 acts on the wire with current I2 is 4∗10^(-6) N.