a circular wire loop of radius 5cm lies in the xy-plane in a magnetic field defined by B=-(4t^2+3t).determine the voltage across the resistor at time t=2s
ε = - dÔ/dt = - d(B•S•cosα)/dt.
cosα = 1
|ε| = S•dB/dt = S•d(4t^2+3t)/dt =
= π•r^2•(8•t+3).
t = 2
|ε| = U = π•0.05^2•(8•2+3)=0.15 V
To determine the voltage across the resistor at time t=2s, we need to calculate the electromotive force (EMF) induced in the circular wire loop.
The EMF induced in a closed loop is defined by Faraday's Law of electromagnetic induction. According to Faraday's Law, the EMF is equal to the negative rate of change of magnetic flux through the loop.
The magnetic flux through a loop is given by the dot product of the magnetic field vector B and the area vector A of the loop. In this case, the loop is a circular wire loop lying in the xy-plane, and the magnetic field is defined as B = -(4t^2 + 3t).
The area vector of a circular loop lying in the xy-plane is perpendicular to the plane of the loop and has a magnitude equal to the product of the radius of the loop and the unit normal vector, n. In this case, the radius of the loop is 5cm, so the area vector is A = πr^2 * n = π(0.05m)^2 * n.
Therefore, the magnetic flux through the loop is given by Φ = B * A = -(4t^2 + 3t) * π(0.05m)^2 * n.
To calculate the rate of change of magnetic flux, we need to differentiate Φ with respect to time t:
(dΦ/dt) = d/dt [-(4t^2 + 3t) * π(0.05m)^2 * n].
(dΦ/dt) = -[8t + 3] * π(0.05m)^2 * n.
Finally, the voltage across the resistor is equal to the EMF induced in the loop:
V = - (dΦ/dt).
Substituting t=2s into the expression for (dΦ/dt), we can calculate the voltage across the resistor at time t=2s.