Final answer:
The magnitude of the magnetic field at point P can be calculated using the rate of change of the electric field and the speed of light.
Step-by-step explanation:
To find the magnitude of the magnetic field at point P, we can use Faraday's law of electromagnetic induction, which relates the rate of change of magnetic flux through a closed loop to the electromotive force (EMF) induced in the loop. The formula is given by:
E=− dΦ/dt
where:
E is the induced EMF,
Φ is the magnetic flux.
The magnetic flux (Φ) through the circular loop is given by:
Φ=B⋅A
where:
B is the magnetic field,
A is the area of the circular loop.
The area A of a circle is given by A=πr2, where r is the radius of the circle.
Given that the radius of the circular area (r) is increasing at a rate of 1.00cm/s, we can express this as dr/dt =0.01m/s.
Now, differentiate both sides of the equation Φ=B⋅A with respect to time:
dΦ/dt =B⋅ dA/dt
Substitute the expression for A and the given rate of change of r:
dΦ/dt =B⋅(2πr⋅ dr/dt )
Now, plug this into Faraday's law:
E=−B⋅(2πr⋅ dr/dt)
The induced EMF (E) is also related to the electric field (E) by:
E=B⋅v
where:
v is the velocity of the conducting loop.
In this case, v is the rate at which the radius is increasing, so v= dr/dt
Equating the two expressions for E, we get:
B⋅v=−B⋅(2πr⋅ dr/dt )
Now, solve for the magnetic field (B):
B=−2πr⋅ dr/dt
Now, plug in the given values:
B=−2π⋅(1.00m)⋅(0.01m/s)
Calculate the magnitude of B to find the answer. Note that the negative sign indicates the direction of the induced magnetic field according to Lenz's law.