A strong electromagnet produces a uniform
field of 1.31 T over a cross-sectional area of
0.195 m2. A coil having 245 turns with a total
resistance of 16
is placed around the electro-
magnet, and the current in the electromagnet
is turned off, reaching zero in 19.3 ms.
What current is induced in the coil?
Answer in units of A.
To find the current induced in the coil, we can use Faraday's law of electromagnetic induction. According to the law, the induced electromotive force (EMF) is equal to the rate of change of magnetic flux.
The formula for the induced EMF is:
EMF = -N * dΦ/dt
Where:
- EMF is the induced electromotive force,
- N is the number of turns in the coil,
- dΦ/dt is the rate of change of magnetic flux.
In this case, when the current in the electromagnet is turned off, the magnetic flux through the coil will change. Since the magnetic field is uniform, the rate of change of magnetic flux can be calculated using the formula:
dΦ/dt = B * A
Where:
- B is the magnetic field,
- A is the cross-sectional area.
Let's calculate the rate of change of magnetic flux:
dΦ/dt = 1.31 T * 0.195 m^2
dΦ/dt = 0.25485 T m^2/s
Now we can substitute this value into the formula for the induced EMF:
EMF = -N * dΦ/dt
EMF = -245 * 0.25485 T m^2/s
EMF = -62.43675 T m^2/s
Since the resistance of the coil is given as 16 Ω, we can use Ohm's law to find the current induced in the coil:
EMF = I * R
Where:
- I is the current induced in the coil,
- R is the resistance of the coil.
Rearranging the formula for I:
I = EMF / R
I = -62.43675 T m^2/s / 16 Ω
We need to convert T m^2/s to A by using the definition of Tesla (T) and Ampere (A):
1 T = 1 N/(A m)
1 T m^2/s = 1 N/(A m) * m^2/s
1 T m^2/s = 1 N/A * m
Substituting the conversion factor:
I = -62.43675 N/A * m / 16 Ω
I = -3.9023 A
The current induced in the coil is approximately -3.9023 A.
To find the current induced in the coil, we can use Faraday's Law of electromagnetic induction, which states that the induced electromotive force (EMF) in a closed loop is equal to the rate of change of magnetic flux through the loop.
The formula for Faraday's Law is:
EMF = -N * ΔΦ/Δt
Where:
- EMF is the induced electromotive force
- N is the number of turns in the coil
- ΔΦ is the change in magnetic flux
- Δt is the change in time
In this case, the magnetic field is uniform, so ΔΦ = B * A, where B is the magnetic field strength (1.31 T) and A is the cross-sectional area of the coil (0.195 m^2).
Substituting the given values, we have:
ΔΦ = (1.31 T) * (0.195 m^2)
To get the change in time, we need to convert the given time of 19.3 ms to seconds by dividing it by 1000:
Δt = 19.3 ms / 1000 = 0.0193 s
Now we can calculate the induced EMF:
EMF = -(245 turns) * [(1.31 T) * (0.195 m^2)] / (0.0193 s)
Simplifying the equation:
EMF = -9.813 A (rounded to three decimal places)
Since the EMF is induced due to the change in magnetic field, it will also induce a current in the coil. The current can be calculated using Ohm's Law:
EMF = I * R
Where:
- I is the induced current in the coil
- R is the total resistance of the coil (16 Ω)
Solving for I:
I = EMF / R
Substituting the values:
I = (-9.813 A) / (16 Ω)
Calculating the result:
I ≈ -0.613 A (rounded to three decimal places)
Therefore, the current induced in the coil is approximately -0.613 A. The negative sign indicates that the direction of the induced current is opposite to the original current in the electromagnet.