The magnetic field `B=2t+4t^(2)` (where t = time) is applied perpendicular to the plane of a circular wire of radius r and resistance R. If all the units are in SI the electric charge that flows through the circular wire during t = 0 s to t = 2 s is

Options

A. `(6pir^(2))/R`
B. `(20pir^(2))/R`
C. `(32pir^(2))/R`
D. `(48pir^(2))/R`

Explanation

To solve the problem, we need to determine the electric charge that flows through a circular wire when a time-dependent magnetic field is applied perpendicular to the plane of the wire. We will follow these steps:

### Step 1: Determine the Magnetic Field
The magnetic field is given as:
\[ B(t) = 2t + 4t^2 \]

### Step 2: Calculate the Magnetic Flux
The magnetic flux (\( \Phi \)) through the circular wire is given by the product of the magnetic field and the area of the circular wire. The area \( A \) of a circle is:
\[ A = \pi r^2 \]
Thus, the magnetic flux at any time \( t \) is:
\[ \Phi(t) = B(t) \cdot A = (2t + 4t^2) \cdot (\pi r^2) \]

### Step 3: Find the Change in Magnetic Flux
To find the change in magnetic flux from \( t = 0 \) to \( t = 2 \):
1. Calculate \( \Phi(2) \):
\[ \Phi(2) = (2(2) + 4(2^2)) \cdot (\pi r^2) = (4 + 16) \cdot (\pi r^2) = 20\pi r^2 \]
2. Calculate \( \Phi(0) \):
\[ \Phi(0) = (2(0) + 4(0^2)) \cdot (\pi r^2) = 0 \]
3. The change in magnetic flux (\( \Delta \Phi \)) is:
\[ \Delta \Phi = \Phi(2) - \Phi(0) = 20\pi r^2 - 0 = 20\pi r^2 \]

### Step 4: Calculate the Induced EMF
According to Faraday's law of electromagnetic induction, the induced electromotive force (EMF, \( \mathcal{E} \)) is given by the negative rate of change of magnetic flux:
\[ \mathcal{E} = -\frac{d\Phi}{dt} \]
Over the interval from \( t = 0 \) to \( t = 2 \), we can calculate the average induced EMF:
\[ \mathcal{E} = -\frac{\Delta \Phi}{\Delta t} = -\frac{20\pi r^2}{2 - 0} = -10\pi r^2 \]

### Step 5: Calculate the Current
Using Ohm's law, the current \( I \) through the wire is given by:
\[ I = \frac{\mathcal{E}}{R} = \frac{-10\pi r^2}{R} \]

### Step 6: Calculate the Total Charge
The total charge \( Q \) that flows through the wire can be calculated by integrating the current over time:
\[ Q = I \cdot t = \left(\frac{-10\pi r^2}{R}\right) \cdot (2) = \frac{-20\pi r^2}{R} \]

Since charge cannot be negative, we take the absolute value:
\[ Q = \frac{20\pi r^2}{R} \]

### Final Answer
The electric charge that flows through the circular wire during \( t = 0 \) s to \( t = 2 \) s is:
\[ Q = \frac{20\pi r^2}{R} \]

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