At time $$t = 0$$ magnetic field of 1000 Gauss is passing perpendicularly through the area defined by the closed loop shown in the figure. If the magnetic field reduces linearly to 500 Gauss, in the next 5s, then induced EMF in the loop is:

Minimum Required Theory 

• Faraday’s Law of Electromagnetic Induction: The magnitude of the induced electromotive force ($$\varepsilon$$) in a closed loop equals the rate of change of magnetic flux ($$\Phi$$) through the loop:
• Unit Conversions:
  • o Area: $$1\text{ cm}^2 = 10^{-4}\text{ m}^2$$
• o Magnetic Field: $$1\text{ Gauss (G)} = 10^{-4}\text{ Tesla (T)}$$
• o Induced EMF: $$1\text{ }\mu\text{V} = 10^{-6}\text{ V}$$
• Area of large rectangle $$= 16\text{ cm} \times 4\text{ cm} = 64\text{ cm}^2$$
• Area of inner cut-out $$= 4\text{ cm} \times 2\text{ cm} = 8\text{ cm}^2$$

$$\varepsilon = \frac{\Delta \Phi}{\Delta t} = A \frac{\Delta B}{\Delta t}$$

Step-by-Step Solution

Step 1: Calculate the effective area ($$A$$) of the closed loop

The geometry consists of a large main rectangle from which a smaller inner rectangular section has been subtracted (or indented).

$$\text{Net Area } A = 64\text{ cm}^2 - 8\text{ cm}^2 = 56\text{ cm}^2$$

Convert the area to SI units ($$\text{m}^2$$):

$$A = 56 \times 10^{-4}\text{ m}^2$$

Step 2: Determine the rate of change of the magnetic field ($$\frac{\Delta B}{\Delta t}$$)

The magnetic field decreases linearly from $$1000\text{ G}$$ to $$500\text{ G}$$ over a time interval $$\Delta t = 5\text{ s}$$.

$$\Delta B = 1000\text{ G} - 500\text{ G} = 500\text{ G}$$

Convert $$\Delta B$$ to Tesla (T):

$$\Delta B = 500 \times 10^{-4}\text{ T}$$

Now, calculate the rate of change:

$$\frac{\Delta B}{\Delta t} = \frac{500 \times 10^{-4}\text{ T}}{5\text{ s}} = 100 \times 10^{-4}\text{ T/s}$$

Step 3: Compute the induced EMF ($$\varepsilon$$)

Substitute the values into Faraday's formula:

$$\varepsilon = A \times \frac{\Delta B}{\Delta t}$$

$$\varepsilon = (56 \times 10^{-4}\text{ m}^2) \times (100 \times 10^{-4}\text{ T/s})$$

$$\varepsilon = 5600 \times 10^{-8}\text{ V}$$

$$\varepsilon = 56 \times 10^{-6}\text{ V}$$

Step 4: Convert to microvolts ($$\mu\text{V}$$)

Since $$10^{-6}\text{ V} = 1\text{ }\mu\text{V}$$:

$$\varepsilon = 56\text{ }\mu\text{V}$$

Final Answer

The induced EMF in the loop is $$56\text{ }\mu\text{V}$$.