A wire of thickness d = 5 mm is tightly wound 200 times around a cylindrical core to form a solenoid. A current I = 0.1 A is sent through the wire. What is the magnetic field on the axis of the solenoid?
2.5*10^(-5) T
To find the magnetic field on the axis of the solenoid, we can use Ampere's Law. Ampere's Law states that the magnetic field inside a solenoid is directly proportional to the current passing through it and the number of loops per unit length.
The formula for the magnetic field inside a solenoid is given by:
B = μ₀ * n * I
Where:
B is the magnetic field,
μ₀ is the permeability of free space (μ₀ ≈ 4π * 10^-7 T·m/A),
n is the number of loops per unit length, and
I is the current passing through the wire.
In this case, the wire is tightly wound around the cylindrical core to form the solenoid. So, the number of loops per unit length (n), also known as the "turns density," is equal to the total number of loops divided by the length of the solenoid.
Given:
Thickness of the wire (d) = 5 mm = 0.005 m
Number of loops around the core (N) = 200
Current passing through the wire (I) = 0.1 A
To find n, we need to determine the length of the solenoid. Since no information about the length is provided in the question, we'll assume a length (L) for the solenoid.
Now, n = N / L
Once we determine n, we can plug in the values into the formula B = μ₀ * n * I to calculate the magnetic field on the axis of the solenoid.
Note: The magnetic field inside the solenoid will be uniform along its axis.
Let's proceed with the calculations.
To find the magnetic field on the axis of the solenoid, we can use the formula:
B = μ₀ * n * I
where B is the magnetic field, μ₀ is the permeability of free space (4π * 10⁻⁷ T m/A), n is the number of turns per unit length, and I is the current.
First, we need to find the number of turns per unit length (n):
n = N / L
where N is the total number of turns and L is the length of the solenoid.
Given that the wire is tightly wound 200 times around the core, we have N = 200.
The length of the solenoid can be calculated using the formula:
L = π * d * N
where d is the thickness of the wire and N is the total number of turns.
Substituting the values, we have:
L = π * 0.005 m * 200
Calculating this, we get:
L = 3.14 m
Now we can find the number of turns per unit length (n):
n = 200 / 3.14
Calculating this, we get:
n ≈ 63.694 m⁻¹
Finally, we can calculate the magnetic field (B):
B = μ₀ * n * I
Substituting the values, we have:
B = 4π * 10⁻⁷ T m/A * 63.694 m⁻¹ * 0.1 A
Calculating this, we get:
B ≈ 0.0796 T
Therefore, the magnetic field on the axis of the solenoid is approximately 0.0796 T.