The figure shows an arbitrarily shaped planar conducting loop A in the XY plane. Two nonintersecting regions with areas $a_1$ and $a_2$ within the loop are subjected to magnetic fields $\vec{B}_1 = \frac{m}{\sqrt{2}} \sin(\omega t) (1\hat{x} + 0\hat{y} + 1\hat{z})$, and $\vec{B}_2 = -\frac{n}{\sqrt{2}} \cos(2\omega t + \pi/4) (0\hat{x} + 1\hat{y} + 1\hat{z})$, respectively. What is the expression for the induced rms voltage in loop A?
To determine the induced rms voltage in the loop A, we need to calculate the emf induced by the changing magnetic fields in the two areas, \(a_1\) and \(a_2\), and then find the rms value.
Concepts Involved
- Faraday's Law of Electromagnetic Induction, which states that the induced emf is equal to the negative rate of change of magnetic flux through the loop.
- Root mean square (rms) voltage, which for sinusoidal waveforms is given by \(V_{\text{rms}} = \frac{V_0}{\sqrt{2}}\).
Step-by-step Solution
- Calculate the magnetic flux \(\Phi_1\) through area \(a_1\) due to \(\vec{B}_1\):
- Magnetic flux: \(\Phi_1 = a_1 \times \left(\frac{m}{\sqrt{2}} \sin(\omega t)\right)\)
- Calculate the induced emf \(\varepsilon_1\) in this area using Faraday’s Law:
- \(\varepsilon_1 = -\frac{d\Phi_1}{dt} = -a_1 \frac{d}{dt}\left(\frac{m}{\sqrt{2}} \sin(\omega t)\right)\)
- Calculating the derivative: \(\varepsilon_1 = -a_1 \times \frac{m}{\sqrt{2}} \omega \cos(\omega t)\)
- Calculate the magnetic flux \(\Phi_2\) through area \(a_2\) due to \(\vec{B}_2\):
- Magnetic flux: \(\Phi_2 = a_2 \times \left(-\frac{n}{\sqrt{2}} \cos(2\omega t + \pi/4)\right)\)
- Calculate the induced emf \(\varepsilon_2\) in this area using Faraday’s Law:
- \(\varepsilon_2 = -\frac{d\Phi_2}{dt} = a_2 \frac{d}{dt}\left(\frac{n}{\sqrt{2}} \cos(2\omega t + \pi/4)\right)\)
- Calculating the derivative: \(\varepsilon_2 = a_2 \times \frac{n}{\sqrt{2}} \times 2\omega \sin(2\omega t + \pi/4)\)
- The total induced emf \(\varepsilon\) in the loop is the sum of \(\varepsilon_1\) and \(\varepsilon_2\):
- \(\varepsilon = \varepsilon_1 + \varepsilon_2\)
- Calculate the rms value of the total induced emf:
- The rms voltage expression is: \(V_{\text{rms}} = \sqrt{\frac{a_1^2 \omega^2 m^2 + 4a_2^2 \omega^2 n^2}{4}}\). This is derived from the root mean square formula considering both contributions independently and then summing them in quadrature.
Conclusion
The expression for the induced rms voltage in loop A is \(\sqrt{\frac{a_1^2 \omega^2 m^2 + 4a_2^2 \omega^2 n^2}{4}}\).
Important Questions from Faraday's Law
In the table shown, List I and List II, respectively, contain terms appearing on the left-hand side and the right-hand side of Maxwell’s equations (in their standard form). Match the left-hand side with the corresponding right-hand side.
List I
List II
1.
∇. D
P
0
2.
∇ × E
Q
\(\rho_v\)
3.
∇. B
R
\( - \frac{{\partial B}}{{\partial t}}\)
4.
∇ × H
S
\(J+ \frac{{\partial D}}{{\partial t}}\)
“By the motion of the conductor or the coil in a magnetic field, i.e., the magnetic field is stationary and the moving conductors cut through it. The EMF generated in this way is normally called dynamically induced EMF.”
The given statement is specified by which of the following laws?
A flux of 0.25 mWb is produced by a coil of 1000 turns wound on a ring with a current of 2 A in it. Calculate the e.m.f induced in the coil when a current of 10 A is switched off, assuming the current will fall to zero in 1 millisecond.
Which of the following law states that “whenever the magnetic flux linked with a conductor or coil changes, an emf is induced in it?
All induction type transducers are based on
