A 2.00 cm x 2.00 cm square loop of wire with resistance 1.30×10^−2 Ohms is parallel to a long straight wire. The near edge of the loop is 1.00 cm from the wire. The current in the wire is increasing at the rate of 110 A/s. What is the current in the loop?
You have to find the flux, and since it is not constant with distance, integrate BdA outward
EMF= d/dt flux= d/dt INT BdA =d/dt INT .02B(r)dr from .01 to .03m outward. You know B as a function of r. Then, take the d/dt of B which is a function of current.
Ok, so I did INT from .01 to .03 of mu_0*I/r which ended up being mu_0*I*(ln .03-1n .01). I'm confused why you only used .02 as the area. It doesn't give radius of the wire and area of the wire is .02^2. When taking the derivative then, d/dt of I is just dI/dt? so i should just substitute in the 110 A/s? Also, do you do anything with the resistance or is that just extra information?
To find the current in the loop, we can use Faraday's law of electromagnetic induction. According to the law, the induced electromotive force (EMF) in a closed loop 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 (B) and the area (A) of the loop.
Step 1: Calculate the magnetic field strength (B)
The magnetic field strength is given by the formula:
B = μ₀ * I / (2 * pi * r)
where μ₀ is the permeability of free space (4 * pi * 10^(-7) T*m/A), I is the current in the wire, and r is the distance between the wire and the loop.
In this case, the distance between the wire and the loop is 1.00 cm = 0.01 m.
B = (4 * pi * 10^(-7) T*m/A) * I / (2 * pi * 0.01 m)
B = 2 * 10^(-5) T * I
Step 2: Calculate the area (A) of the loop
The area of the loop is given by the formula:
A = length * width
In this case, the length and width of the loop are both 2.00 cm = 0.02 m.
A = 0.02 m * 0.02 m
A = 4 * 10^(-4) m^2
Step 3: Calculate the rate of change of magnetic flux
The rate of change of magnetic flux is given by the formula:
dΦ/dt = B * dA/dt
Since the area of the loop is not changing, dA/dt = 0.
dΦ/dt = B * dA/dt
dΦ/dt = 2 * 10^(-5) T * I * 0
Therefore, the rate of change of magnetic flux is zero.
Step 4: Calculate the induced EMF
According to Faraday's law, the induced EMF is equal to the rate of change of magnetic flux.
ΔV = dΦ/dt
Since the rate of change of magnetic flux is zero, the induced EMF is also zero.
Step 5: Calculate the current in the loop
The current in the loop is given by Ohm's law:
I_loop = ΔV / R
where R is the resistance of the loop.
In this case, the resistance of the loop is 1.30 × 10^(-2) Ω.
I_loop = 0 / (1.30 × 10^(-2) Ω)
I_loop = 0 A
Therefore, the current in the loop is zero.
To find the current in the loop, we can use Faraday's Law of electromagnetic induction. According to Faraday's Law, the induced electromotive force (EMF) in a closed loop is equal to the rate of change of the magnetic flux passing through the loop. This can be expressed as:
EMF = -d(Φ)/dt
Where EMF is the electromotive force, Φ is the magnetic flux, and dt is the change in time.
In this case, the magnetic field generated by the long straight wire induces a magnetic flux in the loop. The magnetic flux can be calculated using the formula:
Φ = B * A
Where B is the magnetic field and A is the area of the loop.
Since the magnetic field is perpendicular to the loop, we can use Ampere's Law to calculate it:
B = (μ₀ * I)/ (2π * r)
Where μ₀ is the permeability of free space, I is the current in the wire, and r is the distance from the wire to the near edge of the loop.
Plugging this value of B into the formula for magnetic flux, we get:
Φ = [(μ₀ * I) / (2π * r)] * A
Now, we can differentiate both sides of the equation with respect to time to find the rate of change of the magnetic flux:
d(Φ)/dt = [(μ₀ * dI/dt) / (2π * r)] * A
We know the values of dI/dt, r, and A, so we can substitute them into the equation to calculate the rate of change of the magnetic flux. Finally, we can substitute this value into Faraday's Law to find the induced electromotive force (EMF) in the loop.
Since the loop has resistance, we can apply Ohm's Law to find the current in the loop:
EMF = I_loop * R_loop
Rearranging the equation, we can solve for the current in the loop:
I_loop = EMF / R_loop
By substituting the values of EMF and R_loop into the equation, we can calculate the current in the loop.