During a 74-ms interval, a change in the current in a primary coil occurs. This change leads to the appearance of a 4.6-mA current in a nearby secondary coil. The secondary coil is part of a circuit in which the resistance is 12 Ù. The mutual inductance between the two coils is 3.2 mH. What is the change in the primary current?
The voltage in the second coil is
V2 = I2*R2 = 4.6*10^-3*12 = 5.52*10^-2 V
This equals M12*dI1/(dt) caused by mutual inductance M12
M12 is the mutual inductance and dt = 74*10^-3 s.
Solve for the current change dI1
dI1 = (5.52*10^-2 V)*74*10^-3s/(3.2*10^-3 H) = 1.27 Amp
To find the change in the primary current, we can use Faraday's law of electromagnetic induction.
According to Faraday's law, the induced voltage in the secondary coil is given by the formula:
V = -M * (dI/dt)
Where V is the induced voltage, M is the mutual inductance between the two coils, and (dI/dt) is the rate of change of current in the primary coil.
Given:
Mutual inductance (M) = 3.2 mH = 3.2 * 10^(-3) H
Induced current in the secondary coil (I) = 4.6 mA = 4.6 * 10^(-3) A
Resistance (R) = 12 Ω
We can rearrange the formula to solve for the rate of change of current:
(dI/dt) = -V / M
Substituting the given values, we have:
(dI/dt) = -(4.6 * 10^(-3) A) / (3.2 * 10^(-3) H)
(dI/dt) = -1.4375 A/s
Now, to find the change in the primary current, we can use Ohm's Law:
V = I * R
Rearranging the formula, we have:
ΔI = ΔV / R
Where ΔI is the change in the primary current and ΔV is the change in voltage across the resistance.
To find ΔV, we can use the formula:
ΔV = (dI/dt) * R
Substituting the values we have:
ΔV = (-1.4375 A/s) * 12 Ω
ΔV = -17.25 V
Finally, we can find the change in the primary current:
ΔI = (-17.25 V) / 12 Ω
ΔI = -1.4375 A
Therefore, the change in the primary current is approximately -1.4375 A.
To find the change in the primary current, we can use Faraday's Law of electromagnetic induction, which states that the induced electromotive force (emf) in a circuit is equal to the rate of change of magnetic flux through the circuit.
The induced emf in the secondary coil can be calculated using the formula:
emf = -M * (dI2/dt)
Where:
- emf is the induced electromotive force,
- M is the mutual inductance between the two coils, and
- dI2/dt is the rate of change of current in the secondary coil.
Given:
- emf = 4.6 mA = 4.6 * 10^(-3) A,
- M = 3.2 mH = 3.2 * 10^(-3) H.
Now, let's rearrange the formula to solve for dI2/dt:
dI2/dt = -(emf / M)
Substituting the given values:
dI2/dt = -(4.6 * 10^(-3) A) / (3.2 * 10^(-3) H)
Simplifying:
dI2/dt = -1.4375 A/s
Since the primary and secondary currents are related through the mutual inductance, we can assume that the same rate of change applies to the primary current as well.
Therefore, the change in the primary current is 1.4375 A/s.