A piece of copper wire has a resistance per unit length of 5.50 x 10^-3 Ω/m. The wire is wound into a thin, flat coil of many turns that has a radius of 0.200 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.
Magnetic field at the center of a circular current-carrying loop of N turns and radius r is
B=Nμ₀I/2r….. (1)
N=L/2πr ….(2)
I=U/R=U/R₀L ….(3)
Substitution (2) and (3) in (1) gives
B=Nμ₀I/2r = (L/2πr) •(μ₀/2r) •(U/R₀L) = μ₀U/4πr²R₀=
=4π•10⁻⁷•12/ 4π •(0.2)² •5.5•10⁻³ =5.45•10⁻³ T
To find the magnetic field strength at the center of the coil, we can use Ampere's law. Ampere's law states that the magnetic field along a closed loop is proportional to the current passing through the loop.
First, let's find the total current passing through the coil. The current is determined by the voltage and the resistance of the wire.
The resistance per unit length is given as 5.50 x 10^-3 Ω/m. Since the wire forms a coil of many turns, the total length of the wire can be found using the formula for the circumference of a circle, which is 2πr, where r is the radius.
The circumference of the coil is 2π(0.200 m) = 1.26 m.
We can then find the total resistance of the wire using the formula: Resistance = (resistance per unit length) x (length of the wire).
Resistance = (5.50 x 10^-3 Ω/m) x (1.26 m) = 6.93 x 10^-3 Ω.
Now, we can use Ohm's law to find the current passing through the coil. Ohm's law states that current is equal to voltage divided by resistance.
Current = Voltage / Resistance = 12.0 V / 6.93 x 10^-3 Ω = 1730 A.
Now that we know the current passing through the coil, we can use Ampere's law to find the magnetic field strength at the center of the coil. Ampere's law states that the magnetic field strength along a closed loop is equal to the product of the permeability of free space (μ₀) and the current passing through the loop, divided by 2π times the distance from the wire.
The formula is: Magnetic field strength = (μ₀ x Current) / (2π x Distance).
The distance from the wire to the center of the coil is equal to the radius of the coil, which is 0.200 m.
Plugging in the values, we get:
Magnetic field strength = (4π x 10^-7 T·m/A x 1730 A) / (2π x 0.200 m) = (6.92 x 10^-4 T·m/A) / (0.400 m) = 1.73 x 10^-3 T.
Therefore, the magnetic field strength at the center of the coil is 1.73 x 10^-3 Tesla.