A magnetic field is uniform in a region of radius 10 cm, and increasing at 0.02 Tesla/second.
To calculate the magnitude of the induced electric field in a circular region, you can use Faraday's law of electromagnetic induction. According to Faraday's law, the magnitude of the induced electric field (E) is equal to the rate of change of the magnetic flux (Φ) passing through the circular region. The formula is given as:
E = -dΦ/dt
Where:
E is the magnitude of the induced electric field,
dΦ is the change in the magnetic flux, and
dt is the change in time.
In this case, the magnetic field is said to be uniform and increasing at 0.02 T/s in a circular region of radius 10 cm.
First, let's calculate the magnetic flux (Φ) passing through the circular region. The magnetic flux is given by the formula:
Φ = B * A
Where:
B is the magnetic field strength, and
A is the area of the circular region.
Given that the radius of the circular region is 10 cm, the area can be calculated as:
A = π * (radius)^2
= π * (0.1 m)^2
= 0.0314 m^2
Now, we can calculate the change in magnetic flux (dΦ) over a given time interval.
dΦ = B * dA
Given that the magnetic field is increasing at a rate of 0.02 T/s, and dA = A (since it's a uniform magnetic field), we have:
dΦ = (0.02 T/s) * (0.0314 m^2)
= 0.000628 T·m^2/s
Finally, we can substitute the values into the formula for the magnitude of the induced electric field (E).
E = -dΦ/dt
= -(0.000628 T·m^2/s)/(1 s)
≈ -0.000628 V/m
Note: The negative sign indicates that the direction of the induced electric field is opposite to the rate of change of the magnetic field. This is according to Lenz's law, which states that an induced electric field is produced to oppose the change in the magnetic field.
Therefore, the magnitude of the induced electric field in the circular region is approximately 0.000628 V/m in the opposite direction of the increasing magnetic field.