a) The Earth magnetic field on the equator is directed horizontally from South to North. 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 calculate the total magnetic field strength at a point directly below the power line, we need to consider the magnetic field produced by both the Earth's magnetic field and the current in the power line.
The magnetic field produced by a current-carrying wire can be calculated using the formula:
B = (µ0 * I) / (2π * r)
Where:
B = magnetic field strength
µ0 = permeability of free space (4π * 10^-7 T*m/A)
I = current in the wire (100 A)
r = distance from the wire (20 m)
For the Earth's magnetic field, we can assume it as a constant horizontal magnetic field pointing from South to North.
Therefore, the total magnetic field at the point below the power line will be the vector sum of the Earth's magnetic field and the magnetic field produced by the power line.
Since both magnetic fields are horizontal, the total magnetic field strength will be the square root of the sum of the squares of the individual magnetic field strengths. We can calculate the total magnetic field strength using the formula for vector addition:
B_total = √((B_earth)^2 + (B_power line)^2)
Substitute the values into the formulas to find B_earth and B_power line.
B_earth = 2π * 4π*10^-7
= 8*10^-7 T
B_power line = (4π*10^-7 * 100) / (2π*20)
= 2*10^-6 T
Now, calculate the total magnetic field strength:
B_total = √((8*10^-7)^2 + (2*10^-6)^2)
= √(6.4*10^-13 + 4*10^-12)
= √(4.64*10^-12)
= 6.8*10^-6 T
So, the total magnetic field strength at 20 m directly below the power line is 6.8*10^-6 T. The direction of this magnetic field will be a vector sum of the Earth's magnetic field and the magnetic field produced by the power line.
b) To calculate the force between the two parallel wires, we can use the formula for the force between two parallel current-carrying wires:
F = (µ0 * I1 * I2 * l) / (2π * d)
Where:
F = force
µ0 = permeability of free space (4π * 10^-7 T*m/A)
I1 = current in wire 1 (1 A)
I2 = current in wire 2 (2 A)
l = length of wire 2 (10 m)
d = separation between the wires (10 cm = 0.1 m)
Substitute the values into the formula:
F = (4π * 10^-7 * 1 * 2 * 10) / (2π * 0.1)
= (8π * 10^-6) / (0.2π)
= 40 * 10^-6
= 4*10^-5 N
Therefore, the force with which the wire with current I1 acts on a 10-m long segment of the wire with current I2 is 4*10^-5 N.