A square loop with resistance R=0.001 Ohms,mass 1.0 gram, and side length 1.0 m is falling into a region of uniform magnetic field. The magnetic field has strength 1 Gauss and is pointing out of the page. The speed of the loop
just before it enters the region with the uniform magnetic field is 1.0 m/s.
- As the loop enters the region with non-zero magnetic field, what is the value of the current generated in the
loop?
-Is the current in the loop clockwise or anti-clockwise?
Could someone show me how to approach this problem? I have no idea where to start.
To approach this problem, we can use Faraday's Law of electromagnetic induction, which states that the emf induced in a circuit is proportional to the rate of change of magnetic flux passing through the loop. From the given information, we can determine the induced current as the loop enters the region with a magnetic field.
Step 1: Calculate the magnetic flux passing through the loop
The magnetic flux (Φ) passing through a loop is given by the equation:
Φ = B * A * cos(θ),
where B is the magnetic field strength, A is the area of the loop, and θ is the angle between the magnetic field and the normal to the loop.
In this case, the loop is square, so the area of the loop (A) is given by:
A = (side length)^2.
Given:
B = 1 Gauss (convert to Tesla by dividing by 10,000)
A = (1.0 m)^2
Step 2: Determine the rate of change of the magnetic flux
The rate of change of the magnetic flux (dΦ/dt) is obtained by differentiating the magnetic flux with respect to time. In this problem, the loop is falling into the magnetic field, so the area (A) of the loop is changing with time.
Step 3: Calculate the induced emf
The induced emf (ε) in the loop is given by the equation:
ε = -dΦ/dt,
where -dΦ/dt represents the negative rate of change of the magnetic flux.
Step 4: Calculate the induced current
The induced current (I) in the loop can be calculated using Ohm's Law:
I = ε / R,
where R is the resistance of the loop.
Step 5: Determine the direction of the current
According to Lenz's Law, the direction of the current induced in a loop is such that it opposes the change in magnetic flux that created it. In this problem, the magnetic field is pointing out of the page, and as the loop enters the region, it experiences an increasing magnetic flux. Therefore, the induced current will create a magnetic field pointing into the page to oppose the external magnetic field.
Now, let's perform the calculations.
Step 1: Calculate the magnetic flux passing through the loop
B = 1 Gauss / 10,000 = 0.0001 Tesla
A = (1.0 m)^2
θ = 0° (because the magnetic field is perpendicular to the loop)
Φ = (0.0001 Tesla) * (1.0 m)^2 * cos(0°)
Step 2: Determine the rate of change of the magnetic flux
The loop is falling into the magnetic field, so the area (A) of the loop is changing with time. Let's assume the loop is entering with a constant velocity (v), so the rate of change of the area is given by:
(dA/dt) = -v * (side length)
(dA/dt) = -(1.0 m/s) * (1.0 m)
Step 3: Calculate the induced emf
ε = - (dΦ/dt)
ε = - (d/dt)[(0.0001 Tesla) * (1.0 m)^2 * cos(0°)]
Step 4: Calculate the induced current
I = ε / R
I = [-(d/dt)[(0.0001 Tesla) * (1.0 m)^2 * cos(0°)]] / (0.001 Ohms)
Step 5: Determine the direction of the current
The current in the loop will create a magnetic field pointing into the page to oppose the external magnetic field.
Please note that steps 2 to 4 involve calculus and differentiation. You can use the appropriate differentiation method (e.g., product rule, chain rule) to calculate the values.
To find the value and direction of the current generated in the loop as it enters the region with the magnetic field, you can use Faraday's law of electromagnetic induction.
Faraday's law states that the induced electromotive force (emf) in a circuit is equal to the rate of change of magnetic flux through the circuit. Mathematically, it is expressed as:
emf = -dΦ/dt
Where emf is the induced electromotive force, dΦ is the change in magnetic flux, and dt is the change in time.
In this case, the loop is falling into a region of uniform magnetic field. When the loop enters the magnetic field, the magnetic flux through the loop changes, thereby inducing an emf in the loop.
The magnetic flux (Φ) through the loop is given by the equation:
Φ = B * A
Where B is the magnetic field strength and A is the area of the loop.
In this case, the magnetic field strength (B) is given as 1 Gauss, which is equivalent to 0.0001 Tesla. The area of the loop (A) is given by the equation:
A = l^2
Where l is the length of one side of the loop, which is 1 meter in this case.
Therefore, the magnetic flux through the loop (Φ) can be calculated as:
Φ = B * A
= (0.0001 Tesla) * (1 meter^2)
= 0.0001 Weber
Now, to find the induced emf, we need to calculate the rate of change of magnetic flux. Since the loop is falling into the magnetic field, the change in magnetic flux can be expressed as:
dΦ/dt = (Φ_final - Φ_initial) / t
Where Φ_final is the final magnetic flux, Φ_initial is the initial magnetic flux, and t is the time taken for the change to occur.
At the moment when the loop is just about to enter the region with the magnetic field, the initial magnetic flux is zero. The final magnetic flux is given as:
Φ_final = B * A
= (0.0001 Tesla) * (1 meter^2)
= 0.0001 Weber
The time taken for the loop to enter the magnetic field is not given in the question, so we cannot calculate the exact rate of change of magnetic flux.
To determine the direction of the induced current, we use Lenz's law, which states that the direction of the induced current opposes the change that caused it. In this case, as the loop falls into the magnetic field, the flux through the loop increases. According to Lenz's law, the induced current will create a magnetic field that opposes the increase in flux.
Since the magnetic field is pointing out of the page, to oppose it, the induced current in the loop would create a magnetic field pointing into the page. This means that the current in the loop will be anti-clockwise.