Problem Statement:
A rectangular coil with 100 turns, each of length 1 cm and breadth 0.5 cm, is suspended in a radial magnetic field with an induction of 0.002 Tesla. The torsional constant of the suspension is 2 × 10^-8 Nm/degree. We need to calculate the sensitivity of the microammeter connected to the coil.

Solution:
To calculate the sensitivity of the microammeter, we need to determine the change in the magnetic field strength (B) produced by a change in current (I) through the coil. This change in magnetic field strength will cause a torsional force on the coil, resulting in a deflection that can be measured by the microammeter.

Step 1: Calculate the Magnetic Moment (µ) of the Coil:
The magnetic moment of a coil is given by the product of the number of turns (N), the current (I) passing through the coil, and the area (A) of the coil. The area of the coil can be calculated by multiplying the length (l) and breadth (b) of the coil.

Area of the coil (A) = length (l) × breadth (b) = 1 cm × 0.5 cm = 0.5 cm^2

Number of turns (N) = 100

Magnetic moment (µ) = NIA

Step 2: Calculate the Change in Magnetic Field Strength (ΔB):
The change in magnetic field strength is directly proportional to the change in current passing through the coil.

ΔB = µ0 × ΔI / (2πr^3)

Where µ0 is the permeability of free space (4π × 10^-7 Tm/A) and r is the radius of the coil.

Step 3: Calculate the Torsional Force (τ):
The torsional force exerted on the coil is given by the equation:

τ = -kθ

Where k is the torsional constant of the suspension and θ is the angle of deflection.

Step 4: Calculate the Sensitivity (S):
The sensitivity of the microammeter is given by the equation:

S = ΔB / τ

Substituting the values from the previous steps, we can calculate the sensitivity.

Summary:
The sensitivity of the microammeter connected to the rectangular coil can be calculated by determining the change in magnetic field strength produced by a change in current through the coil. This change in magnetic field strength causes a torsional force on the coil, resulting in a deflection that can be measured by the microammeter. By following the steps outlined above, we can calculate the sensitivity of the microammeter.