What current is required in the winding of a long solenoid that has 1000 turns
uniformly distributed over a length of 0.4 m, to produce at the center of the solenoid
a magnetic field of magnitude 1.0 x 10
-4
T?
To calculate the current required in the winding of a long solenoid, we can use the formula for the magnetic field inside a solenoid:
B = μ₀ * n * I
Where:
B is the magnetic field (given as 1.0 x 10^-4 T)
μ₀ is the permeability of free space (μ₀ ≈ 4π × 10^-7 T m/A)
n is the number of turns per unit length (n = N / L, where N is the total number of turns and L is the length of the solenoid)
I is the current in the winding
First, let's calculate the number of turns per unit length:
n = 1000 turns / 0.4 m = 2500 turns/m
Now, substitute the values into the formula:
1.0 x 10^-4 T = (4π × 10^-7 T m/A) * (2500 turns/m) * I
Next, solve for I:
I = 1.0 x 10^-4 T / [(4π × 10^-7 T m/A) * (2500 turns/m)]
I ≈ 0.4 A
Therefore, a current of approximately 0.4 A is required in the winding of the solenoid to produce a magnetic field of magnitude 1.0 x 10^-4 T at the center of the solenoid.