A circular ring of diameter 20cm has a resistance `0.01Omega` How much charge will flow through the ring if it is rotated from positon perpendicular to the uniform magnetic field of B=2T to a position parallel to field?

Options

A. 4 C
B. 6.28 C
C. 3.14 C
D. 2.5 C

Explanation

To solve the problem, we need to calculate the charge that flows through a circular ring when it is rotated from a position perpendicular to a magnetic field to a position parallel to the field. Here’s a step-by-step solution:

### Step 1: Determine the area of the circular ring
The area \( A \) of a circular ring can be calculated using the formula:
\[ A = \pi r^2 \]
where \( r \) is the radius of the ring. Given that the diameter is 20 cm, the radius \( r \) is:
\[ r = \frac{20 \, \text{cm}}{2} = 10 \, \text{cm} = 0.1 \, \text{m} \]
Now, substituting the radius into the area formula:
\[ A = \pi (0.1)^2 = \pi (0.01) = 0.01\pi \, \text{m}^2 \]

### Step 2: Calculate the initial magnetic flux
The initial magnetic flux \( \Phi_i \) when the ring is perpendicular to the magnetic field is given by:
\[ \Phi_i = B \cdot A \cdot \cos(0^\circ) = B \cdot A \]
Substituting the values:
\[ B = 2 \, \text{T}, \quad A = 0.01\pi \, \text{m}^2 \]
Thus,
\[ \Phi_i = 2 \cdot (0.01\pi) = 0.02\pi \, \text{Wb} \]

### Step 3: Calculate the final magnetic flux
When the ring is parallel to the magnetic field, the angle between the magnetic field and the area vector is 90 degrees:
\[ \Phi_f = B \cdot A \cdot \cos(90^\circ) = 0 \]

### Step 4: Calculate the change in magnetic flux
The change in magnetic flux \( \Delta \Phi \) is:
\[ \Delta \Phi = \Phi_f - \Phi_i = 0 - 0.02\pi = -0.02\pi \, \text{Wb} \]

### Step 5: Calculate the induced EMF
According to Faraday's law of electromagnetic induction, the induced EMF \( \mathcal{E} \) is given by:
\[ \mathcal{E} = -\frac{\Delta \Phi}{\Delta t} \]
Assuming the time taken for the rotation is \( t \), we have:
\[ \mathcal{E} = \frac{0.02\pi}{t} \, \text{V} \]

### Step 6: Calculate the induced current
Using Ohm's law, the induced current \( I \) can be calculated as:
\[ I = \frac{\mathcal{E}}{R} \]
where \( R = 0.01 \, \Omega \):
\[ I = \frac{0.02\pi/t}{0.01} = \frac{2\pi}{t} \, \text{A} \]

### Step 7: Calculate the charge that flows
The total charge \( Q \) that flows through the ring can be calculated using the relationship:
\[ Q = I \cdot t \]
Substituting the expression for current:
\[ Q = \left(\frac{2\pi}{t}\right) \cdot t = 2\pi \, \text{C} \]

### Final Answer
Thus, the total charge that flows through the ring is:
\[ Q = 2\pi \, \text{C} \approx 6.28 \, \text{C} \]