Two resistors are connected in a circuit loop of area $5 \ m^2$, as shown in the figure below. The circuit loop is placed on the x-y plane.
When a time-varying magnetic flux, with flux-density $B(t) = 0.5t$ (in Tesla), is applied along the positive z-axis, the magnitude of current $I$ (in Amperes, rounded off to two decimal places) in the loop is __________.
The problem asks for the magnitude of the current induced in a circuit loop due to a time-varying magnetic field, based on Faraday's Law of Induction.
1. Calculate Induced Electromotive Force ($\mathcal{E}$)
The induced EMF ($\mathcal{E}$) is given by Faraday's Law:
$$\mathcal{E} = - \frac{d\Phi_B}{dt}$$
Where $\Phi_B$ is the magnetic flux through the loop, $\Phi_B = \vec{B} \cdot \vec{A}$. Since the loop lies in the x-y plane and the magnetic field $\vec{B}$ is along the positive z-axis, the field is perpendicular to the area vector, and the angle between them is $0^\circ$.
$$\Phi_B = B(t) \cdot A$$
Given parameters:
- Loop Area $A = 5 \text{ m}^2$
- Magnetic Flux Density $B(t) = 0.5t \text{ T}$
Calculate the rate of change of flux ($\frac{d\Phi_B}{dt}$):
$$\frac{d\Phi_B}{dt} = \frac{d}{dt} (B(t) \cdot A) = A \cdot \frac{d}{dt} (0.5t)$$ $$\frac{d\Phi_B}{dt} = 5 \text{ m}^2 \cdot 0.5 \text{ T/s} = 2.5 \text{ V}$$
The induced EMF is:
$$\mathcal{E} = 2.5 \text{ V}$$
2. Calculate the Total Resistance ($R_{\text{total}}$)
The circuit loop contains two $2 \ \Omega$ resistors connected in series (as inferred from the square loop with resistors on two adjacent sides, forming the total resistance of the loop).
$$R_{\text{total}} = 2 \ \Omega + 2 \ \Omega = 4 \ \Omega$$
3. Calculate the Induced Current ($I$)
The magnitude of the induced current $I$ is given by Ohm's law:
$$I = \frac{\mathcal{E}}{R_{\text{total}}}$$ $$I = \frac{2.5 \text{ V}}{4 \ \Omega}$$ $$I = 0.625 \text{ A}$$
4. Final Formatting
Rounding off to two decimal places:
$$I = 0.63 \text{ A}$$
This result falls within the constraint range [0.62, 0.63].
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
