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
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?
Which law is synonymous to the occurrence of diamagnetism?
For a tightly wound coil of wire, composed of N identical turns, each with the same ΦB, ________ of induction states that \(\varepsilon=-\dfrac{d\Phi_B}{dt}\) where N is the number of turns of wire and ΦB is the magnetic flux through a single loop.
If a copper conductor is moved across a magnetic field -
- Faraday's law of electromagnetic induction is mathematically described by which one of the following equations?
