A copper loop and an aluminium loop are removed from a magnetic field in the same time interval. In which loop will the induced current be greater ?

Options

A. Copper loop
B. Aluminium loop
C. Both (a) and (b)
D. None of them

Answer

To determine which loop, between a copper loop and an aluminum loop, will have a greater induced current when both are removed from a magnetic field in the same time interval, we can follow these steps:

### Step 1: Understand the Concept of Induced Current
According to Faraday's law of electromagnetic induction, an electromotive force (EMF) is induced in a loop when there is a change in magnetic flux through the loop. The induced current (I) can be calculated using the formula:
\[ I = \frac{E}{R} \]
where \( E \) is the induced EMF and \( R \) is the resistance of the loop.

### Step 2: Determine the Induced EMF
When both loops are removed from the magnetic field, the change in magnetic flux (\( \Delta \Phi \)) for each loop is the same, as they are both identical in size and are subjected to the same magnetic field. The induced EMF for both loops can be expressed as:
\[ E = -\frac{\Delta \Phi}{\Delta t} \]
Since both loops experience the same change in magnetic flux over the same time interval, the induced EMF will be the same for both loops:
\[ E_{\text{copper}} = E_{\text{aluminum}} \]

### Step 3: Analyze the Resistance of Each Loop
The resistance \( R \) of a loop is given by the formula:
\[ R = \rho \frac{L}{A} \]
where:
- \( \rho \) is the resistivity of the material,
- \( L \) is the length of the wire,
- \( A \) is the cross-sectional area.

Since both loops are identical in size, the length \( L \) and area \( A \) will be the same for both loops. The only difference will be the resistivity \( \rho \) of the materials.

### Step 4: Compare the Resistivities
The resistivity of copper (\( \rho_{\text{copper}} \)) is lower than that of aluminum (\( \rho_{\text{aluminum}} \)):
\[ \rho_{\text{copper}} < \rho_{\text{aluminum}} \]
This means that the resistance of the copper loop will be less than that of the aluminum loop:
\[ R_{\text{copper}} < R_{\text{aluminum}} \]

### Step 5: Calculate the Induced Current
Using the relationship for induced current:
\[ I = \frac{E}{R} \]
Since the induced EMF is the same for both loops and the resistance of the copper loop is less, the induced current in the copper loop will be greater:
\[ I_{\text{copper}} > I_{\text{aluminum}} \]

### Conclusion
Thus, the induced current will be greater in the copper loop compared to the aluminum loop.