To tackle the problem of the induced current in a circular loop of wire placed in a changing magnetic field, we can apply Faraday's law of electromagnetic induction. This law states that a changing magnetic field within a closed loop induces an electromotive force (EMF) in the loop, which in turn generates a current if the circuit is closed. Let's break this down step by step.
Understanding the Magnetic Field Change
The magnetic field is given as \( B(t) = (0.5 \, \text{T}) e^{-2t} \). This indicates that the magnetic field strength decreases exponentially over time. At \( t = 1.0 \, \text{s} \), we can calculate the magnetic field:
- At \( t = 1.0 \, \text{s} \):
- \( B(1) = (0.5 \, \text{T}) e^{-2 \cdot 1} = (0.5 \, \text{T}) e^{-2} \approx 0.5 \times 0.1353 \approx 0.06765 \, \text{T} \)
Determining the Induced Current Direction
To find the direction of the induced current, we can use Lenz's Law, which states that the induced current will flow in a direction that opposes the change in magnetic flux through the loop. Since the magnetic field is decreasing, the induced current will attempt to maintain the magnetic field inside the loop.
Using the right-hand rule, if we point our thumb in the direction of the magnetic field (into the plane of the loop), the fingers curl in the direction of the induced current. Since the magnetic field is decreasing, the induced current will flow in the opposite direction to maintain the field. Therefore, at \( t = 1.0 \, \text{s} \), the induced current will flow in a counterclockwise direction.
Calculating the Induced Current at t = 0s
Next, we need to find the induced current at \( t = 0 \, \text{s} \). First, we calculate the magnetic field at this time:
- At \( t = 0 \, \text{s} \):
- \( B(0) = (0.5 \, \text{T}) e^{-2 \cdot 0} = 0.5 \, \text{T} \)
Now, we can find the change in magnetic flux (\( \Phi \)) through the loop. The area \( A \) of the circular loop can be calculated using the formula \( A = \pi r^2 \), where \( r = 0.1 \, \text{m} \) (10 cm):
- \( A = \pi (0.1)^2 \approx 0.0314 \, \text{m}^2 \)
The magnetic flux \( \Phi \) at \( t = 0 \) is given by:
- \( \Phi(0) = B(0) \cdot A = 0.5 \, \text{T} \cdot 0.0314 \, \text{m}^2 \approx 0.0157 \, \text{Wb} \)
To find the induced EMF (\( \mathcal{E} \)), we need to calculate the rate of change of magnetic flux. The magnetic field at \( t = 0 \) is 0.5 T, and at \( t = 1 \) s, we found it to be approximately 0.06765 T. Thus, the change in flux over the time interval from 0 to 1 second is:
- \( \Delta \Phi = \Phi(1) - \Phi(0) \approx (0.06765 \cdot 0.0314) - 0.0157 \approx 0.00212 - 0.0157 \approx -0.01358 \, \text{Wb} \)
The average induced EMF can be calculated as:
- \( \mathcal{E} = -\frac{\Delta \Phi}{\Delta t} = -\frac{-0.01358}{1} \approx 0.01358 \, \text{V} \)
Finally, we can find the induced current (\( I \)) using Ohm's law, \( I = \frac{\mathcal{E}}{R} \), where \( R = 1092 \, \Omega \):
- \( I = \frac{0.01358}{1092} \approx 0.00001243 \, \text{A} \) or \( 12.43 \, \mu A \)
Summary of Findings
In summary, at \( t = 1.0 \, \text{s} \), the induced current flows in a counterclockwise direction. At \( t = 0 \, \text{s} \), the induced current is approximately \( 12.43 \, \mu A \). This analysis illustrates how changing magnetic fields can induce currents in conductive loops, a fundamental principle in electromagnetism.
