A piece of copper wire has a resistance per unit length of 6.50 10-3 /m. The wire is wound into a thin, flat coil of many turns that has a radius of 0.170 m. The ends of the wire are connected to a 12.0 V battery. Find the magnetic field strength at the center of the coil.
_________ T
B = (4pi x 10^-7)(12.0) / (6.5 x 10^-3)(4pi (0.170^2))
B = 0.000015082 / 0.002360899
B = 6.39 x 10^-3T
B = mu H = (4 pi*10^-7)*i N/(2 r)
N is the number of turns.
r is the coil radius
i is the current
i = V/R
R = N* 2 pi r * (6.5*10^-3 ohm/m)
B = [(4 pi*10^-7)*V*N]/[(N*4*pi*r^2*(6.5*10^-3)]
= 10^-7 V/[r^2*(6.5*10^-3 ohm/m)]
The 4 pi factors and the number of turns N both cancel out. I agree with your formula and answer.
To find the magnetic field strength at the center of the coil, we can use Ampere's Law. Ampere's Law relates the magnetic field around a closed loop to the current passing through that loop and the geometry of the loop itself.
In this case, the coil is wound from a copper wire with a resistance per unit length of 6.50 x 10^-3 Ω/m. The wire is connected to a 12.0 V battery, which produces a current flowing through the wire. However, we do not need to know the actual current value to find the magnetic field strength at the center of the coil.
To solve this problem, we use the formula for the magnetic field of a long, straight wire:
B = (μ0 * I) / (2 * π * r).
Where:
B is the magnetic field strength,
μ0 is the permeability of free space (equal to 4π x 10^-7 Tm/A),
I is the current flowing through the wire, and
r is the distance from the wire.
In this case, we have a coil, so the wire is not straight. However, we can approximate the coil as a circular loop at the center of the coil to use this formula.
The radius of the coil is given as 0.170 m. Plugging in the values, we have:
B = (4π x 10^-7 Tm/A) * (12.0 V) / (6.50 x 10^-3 Ω/m * 4π * (0.170 m)^2).
Simplifying, we get:
B = (4π x 10^-7) * (12.0) / (6.50 x 10^-3 * 4π * 0.170^2).
Cancelling out the common factors, we have:
B = (12.0) / (6.50 x 0.170^2).
Calculating the expression on the right-hand side, we get:
B ≈ 6.39 x 10^-3 T.
So, the magnetic field strength at the center of the coil is approximately 6.39 x 10^-3 Tesla.