A body enters in an MRI machine in `10 s`. If the magnctic field is `1.5 T` and circumference of the MRI mechine is `0.9 m`, then find out the magnitude of emf induced (in milli volt) in the body.

Answer

To find the magnitude of the induced EMF in the body entering the MRI machine, we can follow these steps:

### Step 1: Understand the formula for induced EMF
The induced EMF (ε) can be calculated using Faraday's law of electromagnetic induction, which states:
\[ \epsilon = -\frac{d\Phi}{dt} \]
where \( \Phi \) is the magnetic flux.

### Step 2: Calculate the change in magnetic flux
The magnetic flux \( \Phi \) is given by:
\[ \Phi = B \cdot A \]
where \( B \) is the magnetic field strength (in Tesla) and \( A \) is the area (in square meters) through which the magnetic field lines pass.

### Step 3: Determine the area
Given the circumference of the MRI machine is \( C = 0.9 \, m \), we can find the radius \( r \) using the formula for circumference:
\[ C = 2\pi r \implies r = \frac{C}{2\pi} = \frac{0.9}{2\pi} \]
Now, calculate the area \( A \):
\[ A = \pi r^2 = \pi \left(\frac{0.9}{2\pi}\right)^2 = \frac{0.9^2}{4\pi} = \frac{0.81}{4\pi} \]

### Step 4: Substitute values into the flux equation
Now we can substitute the values of \( B \) and \( A \) into the flux equation:
\[ \Phi = B \cdot A = 1.5 \cdot \frac{0.81}{4\pi} \]

### Step 5: Calculate the change in flux over time
The change in flux over the time interval \( t = 10 \, s \) is:
\[ \Delta \Phi = \Phi - 0 = \Phi \]
Thus, the induced EMF becomes:
\[ \epsilon = \frac{\Delta \Phi}{t} = \frac{1.5 \cdot \frac{0.81}{4\pi}}{10} \]

### Step 6: Simplify the expression
Calculating the above expression:
\[ \epsilon = \frac{1.5 \cdot 0.81}{40\pi} \]
Using \( \pi \approx 3.14 \):
\[ \epsilon = \frac{1.5 \cdot 0.81}{40 \cdot 3.14} \]

### Step 7: Calculate the numerical value
Calculating the numerator:
\[ 1.5 \cdot 0.81 = 1.215 \]
Calculating the denominator:
\[ 40 \cdot 3.14 = 125.6 \]
Now, divide:
\[ \epsilon = \frac{1.215}{125.6} \approx 0.0097 \, V \]

### Step 8: Convert to millivolts
To convert volts to millivolts, multiply by 1000:
\[ \epsilon \approx 0.0097 \times 1000 \approx 9.7 \, mV \]

### Final Answer
The magnitude of the induced EMF in the body is approximately **9.7 mV**.

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