A circular wire is connected to a galvanometer. A solenoid is placed inside the circular wire. The solenoid has 520 turns per meter, a cross-sectional area of 0.007 m^2 and a current charge from 0A to 0.7A in 1.25 ms. The circular wire has a n are of 0.01m^2 and a resistance of 1.6 ohms. What current is displayed on the galvanometer?
To find the current displayed on the galvanometer, we need to analyze the electromagnetic induction caused by the changing current in the solenoid.
The equation that relates the current induced in a loop to the magnetic field and the area of the loop is given by Faraday's Law of Electromagnetic Induction:
ε = -dΦ/dt
Where:
ε represents the induced electromotive force (EMF),
dΦ/dt represents the rate of change of magnetic flux through the loop.
In this case, the circular wire connected to the galvanometer acts as the loop, and the solenoid placed inside it produces the changing magnetic field.
To start, we need to find the rate of change of magnetic flux through the circular wire. The magnetic flux (Φ) is defined as the product of the magnetic field (B) and the area (A) through which it passes:
Φ = B * A
The solenoid has a current that changes from 0A to 0.7A in 1.25 ms. We can calculate the rate of change of current (di/dt) using the formula:
di/dt = Δi/Δt
Given that Δi = 0.7A - 0A = 0.7A
And, Δt = 1.25 ms = 0.00125 s
di/dt = 0.7A / 0.00125s = 560 A/s
Now, we can calculate the magnetic field (B) produced by the solenoid using Ampere's Law:
B = μ₀ * N * I
Where:
μ₀ is the permeability of free space (∼1.26 x 10^-6 T m/A),
N represents the number of turns per unit length of the solenoid (520 turns/m),
I represents the current in the solenoid.
Given that μ₀ = 1.26 x 10^-6 T m/A,
N = 520 turns/m,
I = 0.7A,
B = (1.26 x 10^-6 T m/A) * (520 turns/m) * (0.7A) = 0.36792 T
Now we can substitute the values of B and A into the equation for the rate of change of magnetic flux:
dΦ/dt = B * dA/dt
dA/dt represents the rate of change of the area of the circular wire. However, in this case, the area of the circular wire does not change over time, so dA/dt = 0.
This implies that there is no change in magnetic flux through the circular wire, so the induced EMF (ε) is zero:
ε = -dΦ/dt = 0
Therefore, no current will be displayed on the galvanometer.