The problem involves a long straight wire with a current $I = I_0 \cos(\omega t)$ flowing through it, which generates a time-varying magnetic field. We need to find an approximate expression for the induced current in a rectangular loop of resistance $R$ positioned near this wire. The current $I$ in the wire changes with time, creating a changing magnetic field around it, which induces an EMF (electromotive force) in the loop.

Solution Outline

1. Magnetic Field Due to a Long Straight Wire: The magnetic field $B$ at a distance $r$ from a long straight current-carrying wire is given by: $B = \frac{\mu_0 I}{2 \pi r}$ where $\mu_0$ is the permeability of free space, and $I$ is the current through the wire.

2. Flux Through the Loop: Consider the rectangular loop of height $c$ and width $b$, positioned at a distance $a$ from the wire. The magnetic flux $\Phi$ through the loop depends on the magnetic field, which varies with distance $r$ from the wire. Thus, $\Phi = \int_{a}^{a+b} B \cdot c \, dr = \int_{a}^{a+b} \frac{\mu_0 I c}{2 \pi r} \, dr$ Substituting $I = I_0 \cos(\omega t)$: $\Phi = \int_{a}^{a+b} \frac{\mu_0 I_0 c \cos(\omega t)}{2 \pi r} \, dr$ Evaluating this integral provides the magnetic flux as a function of time.

3. Induced EMF and Current: The induced EMF (Faraday's law) is: $\text{EMF} = -\frac{d\Phi}{dt}$ Differentiating $\Phi$ with respect to $t$ gives the EMF in terms of $\cos(\omega t)$ and $\sin(\omega t)$.

4. Current in the Loop: Using Ohm's law, the induced current $I_{\text{ind}}$ in the loop is: $I_{\text{ind}} = \frac{\text{EMF}}{R}$

5. Direction of EMF: To determine the direction, apply Lenz's Law, which states that the induced current will oppose the change in magnetic flux. The problem specifies that we need to mark the direction on the diagram, which typically would be either clockwise or counterclockwise in the loop.

Would you like a detailed step-by-step calculation for each part, or do you need clarification on any specific step?

Follow-Up Questions

1. How does the distance $a$ from the wire affect the induced current in the loop?
2. What would happen if the current in the wire was constant instead of oscillating?
3. How would the induced current change if the resistance $R$ of the loop increased?
4. What is the role of the loop’s dimensions $b$ and $c$ in calculating the induced EMF?
5. How does the frequency $\omega$ of the oscillating current affect the induced current?

Tip

When dealing with time-varying magnetic fields and induced EMFs, always apply Faraday's and Lenz's laws carefully to determine the magnitude and direction of induced currents.