A flat loop of wire consisting of a single turn of cross-sectional area 8.10 cm2 is perpendicular to a magnetic field that increases uniformly in magnitude from 0.500 T to 1.90 T in 0.95 s. What is the resulting induced current if the loop has a resistance of 2.80 ?
I assume you can find the induced EMF..
E=area*(1.90-.5)/.95
then current=E/resistance
To find the resulting induced current, 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 through the loop is given by the product of the magnetic field strength and the area of the loop:
Φ = B * A
Where:
Φ = magnetic flux
B = magnetic field strength
A = area of the loop
In this case, the area of the loop is given as 8.10 cm^2, which can be converted to m^2:
A = 8.10 cm^2 = 8.10 * 10^(-4) m^2
The initial magnetic field strength is 0.500 T, and the final magnetic field strength is 1.90 T. The rate of change of magnetic field strength can be calculated as:
ΔB = Final B - Initial B = 1.90 T - 0.500 T = 1.40 T
The time taken for this change in magnetic field strength is given as 0.95 s.
Now, we can calculate the rate of change of the magnetic flux:
ΔΦ = ΔB * A / Δt
where Δt = 0.95 s
ΔΦ = (1.40 T) * (8.10 * 10^(-4) m^2) / (0.95 s)
Now, we can calculate the induced electromotive force (EMF) using Faraday's law:
EMF = -dΦ / dt
In our case, the negative sign indicates the direction of the induced current.
Substituting the values:
EMF = -(ΔΦ / Δt)
Now, we have the EMF, and we can use Ohm's law to find the induced current. Ohm's law states that the current is equal to the voltage (EMF) divided by the resistance.
I = EMF / R
Substituting the values:
I = -(EMF) / R
Now, we can substitute the values into the equation and calculate the induced current:
I = -(EMF) / R = -((ΔΦ / Δt) / R)
Calculate the value of ΔΦ / Δt first and then substitute the values:
ΔΦ / Δt = (1.40 T) * (8.10 * 10^(-4) m^2) / (0.95 s)
ΔΦ / Δt = 0.011232 T.m²/s
Now, we can substitute this value into the expression for the current:
I = - (0.011232 T.m²/s) / 2.80 Ω
I = -0.00401 A
Therefore, the resulting induced current in the loop is approximately -0.00401 A.
To find the resulting induced current in the loop, we need to use Faraday's law of electromagnetic induction, which states that the induced electromotive force (EMF) in a loop of wire is equal to the rate of change of magnetic flux through the loop.
The magnetic flux (Φ) through the loop is given by the equation:
Φ = B * A * cos(θ)
Where:
B is the magnetic field strength (in teslas),
A is the cross-sectional area of the loop (in square meters), and
θ is the angle between the magnetic field and the normal to the loop.
In this case, the loop is perpendicular to the magnetic field, so the angle θ is 90 degrees (or π/2 radians). Thus, cos(θ) = 0.
Now, let's calculate the change in magnetic flux (∆Φ) during the time interval (∆t) when the magnetic field changes:
∆Φ = B₂ * A - B₁ * A
Where:
B₁ is the initial magnetic field strength (0.500 T),
B₂ is the final magnetic field strength (1.90 T), and
A is the cross-sectional area of the loop (8.10 cm² = 8.10 × 10^(-4) m²).
∆Φ = (1.90 T) * (8.10 × 10^(-4) m²) - (0.500 T) * (8.10 × 10^(-4) m²)
Now, we need to calculate the average rate of change of magnetic flux (∆Φ/∆t) during the time interval (∆t) when the magnetic field changes:
∆Φ/∆t = (∆Φ) / (∆t)
Where:
∆t is the time interval when the magnetic field changes (0.95 s).
∆Φ/∆t = [(1.90 T) * (8.10 × 10^(-4) m²) - (0.500 T) * (8.10 × 10^(-4) m²)] / (0.95 s)
Now, according to Faraday's law, the induced EMF (ε) in the loop can be calculated as:
ε = -N * (∆Φ/∆t)
Where:
N is the number of turns in the loop (in this case, N = 1, as there is only a single turn).
ε = -1 * [(1.90 T) * (8.10 × 10^(-4) m²) - (0.500 T) * (8.10 × 10^(-4) m²)] / (0.95 s)
Finally, using Ohm's law (V = I * R), we can calculate the induced current (I) in the loop by rearranging the equation:
I = ε / R
Where:
R is the resistance of the wire loop (2.80 Ω).
I = {-1 * [(1.90 T) * (8.10 × 10^(-4) m²) - (0.500 T) * (8.10 × 10^(-4) m²)] / (0.95 s)} / (2.80 Ω)
Evaluating this expression will give us the resulting induced current in the loop.