A single conducting loop of wire has an area of 7.4 10-4 m2 and a resistance of 110 Ω. Perpendicular to the plane of the loop is a magnetic field of strength 0.18 T. At what rate (in T/s) must this field change if the induced current in the loop is to be 0.24 A? I really need some help on this.
Emf=IR=dB/dt=.18/Time
To find the rate at which the magnetic field must change, we can use Faraday's Law of electromagnetic induction. The formula for Faraday's Law is:
EMF (ε) = -N ΔΦ/Δt
Where:
ε = induced electromotive force (EMF)
N = number of turns in the coil
ΔΦ = change in magnetic flux
Δt = change in time
In this case, we have a single conducting loop of wire, so the number of turns (N) is 1. The area of the loop (A) is given as 7.4 x 10^(-4) m^2.
The magnetic flux (Φ) through a loop of area A in a magnetic field B is given by:
Φ = B * A
We are given the magnetic field strength (B) as 0.18 T.
The induced current (I) is given as 0.24 A.
We can rearrange the formula for Faraday's Law to solve for the change in magnetic flux (ΔΦ):
ε = -N ΔΦ/Δt
ΔΦ = -(ε * Δt) / N
Plugging in the values we have:
ΔΦ = - (0.24 A * Δt) / 1
Now let's substitute the expression for magnetic flux (Φ) into our equation:
ΔΦ = - (0.18 T * 7.4 x 10^(-4) m^2)
Now we can solve for the change in time (Δt):
Δt = - ΔΦ / (ε / N)
Plugging in the values:
Δt = - (0.18 T * 7.4 x 10^(-4) m^2) / (0.24 A / 1)
Calculating this expression will give us the change in time (Δt) required for the induced current to be 0.24 A.
To find the rate at which the magnetic field must change, we can use Faraday's law of electromagnetic induction, which states that the induced electromotive force (EMF) is equal to the rate of change of magnetic flux through the loop.
The magnetic flux (Φ) is given by the product of the magnetic field (B), the area of the loop (A), and the cosine of the angle between the magnetic field and the area vector.
Φ = B * A * cos(θ)
In this case, the magnetic field is perpendicular to the plane of the loop, so the angle (θ) between the magnetic field and the area vector is 90 degrees, which means cos(θ) = 0.
Φ = B * A * cos(90°) = B * A * 0 = 0
Since the magnetic flux is zero, we need to find the rate at which it changes (∆Φ/∆t) to induce a current in the loop.
According to Faraday's law, the induced EMF is equal to the negative rate of change of magnetic flux:
EMF = -∆Φ/∆t
We can rearrange this equation to solve for the rate of change of magnetic field (∆B/∆t):
∆B/∆t = -(EMF / A)
Substituting the given values: EMF = 0.24 A and A = 7.4 * 10^-4 m^2
∆B/∆t = -(0.24 A / 7.4 * 10^-4 m^2)
Calculating this equation gives us the rate at which the magnetic field must change, expressed in Teslas per second (T/s).