A circular loop lies with its plane normal to a magnetic field given by B=(0.200 - 0.0200t) T, where t denotes the time in seconds. The induced emf in the loop is 1.36 V. Calculate the radius of the loop.
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physics
To calculate the radius of the loop, we can use Faraday's law of electromagnetic induction. According to Faraday's law, the induced emf (ε) in a conducting loop is given by the equation:
ε = -N(dΦ/dt)
where ε is the induced emf, N is the number of turns in the loop, and dΦ/dt is the rate of change of magnetic flux through the loop.
In this case, we have been given the induced emf (ε) as 1.36 V. In order to calculate the radius of the loop, we need to find the rate of change of magnetic flux (dΦ/dt).
The magnetic flux (Φ) through a loop is given by the equation:
Φ = B⋅A
where B is the magnetic field and A is the area of the loop.
As per the given problem, the magnetic field (B) is given as B = (0.200 - 0.0200t) T.
To find the rate of change of magnetic flux (dΦ/dt), we differentiate the equation for magnetic flux with respect to time (t):
dΦ/dt = d/dt (B⋅A)
Since the magnetic field B is changing with time, we also need to consider the rate of change of B with respect to time:
dΦ/dt = (dB/dt)⋅A
Now we substitute the given equation for B and the unknown value of A into the equation for the rate of change of magnetic flux (dΦ/dt):
dΦ/dt = ((0.0200)⋅A) T/s
Putting everything together, the equation for the induced emf becomes:
ε = -N(dΦ/dt) = -N((0.0200)⋅A) T/s
With the given value of ε as 1.36 V, we can rearrange the equation to solve for the area (A):
A = ε/(-N(0.0200)) m^2
Finally, we can use the formula for the area of a circle, A = πr^2, to find the radius (r) of the loop:
r^2 = A/π
r = sqrt(A/π)
Now that we have all the necessary information, we can substitute the known values into the equation to find the radius of the loop.