Given:
- BC = 4V
- aab = 200 kilo ohm = 200 x 10^3 ohm
- 1 kilo ohm = 1 x 10^3 ohm

To Find:
- Current gain

Explanation:

Step 1: Identify the Circuit Type
The given circuit is a common-emitter configuration of a bipolar junction transistor (BJT).

Step 2: Analyze the Circuit
To find the current gain, we need to analyze the circuit and determine the values of the base current (IB), collector current (IC), and current gain (β).

Step 3: Apply Ohm's Law
Using Ohm's Law, we can determine the values of current and voltage in the circuit.

- In the given circuit, BC = 4V.
- The voltage across the base-emitter junction of the BJT is approximately 0.7V in forward bias.

Therefore, VB = BC + 0.7V = 4V + 0.7V = 4.7V

Step 4: Calculate the Base Current (IB)
Using Ohm's Law (V = IR), we can determine the base current (IB) flowing through the resistor connected to the base terminal.

IB = VB / aab = 4.7V / 200 x 10^3 ohm = 0.0235 mA

Step 5: Calculate the Collector Current (IC)
The collector current (IC) is given by the equation IC = β * IB, where β is the current gain of the transistor.

Step 6: Calculate the Current Gain (β)
The current gain (β) is defined as the ratio of collector current (IC) to base current (IB).

β = IC / IB

Step 7: Substitute Values and Calculate
Substituting the values of IC and IB into the equation, we can calculate the current gain.

β = IC / IB = IC / (VB / aab)

Step 8: Calculate IC
To calculate IC, we need to find the voltage across the collector-emitter junction of the BJT.

VC = BC + VB

Using Ohm's Law, we can determine IC.

IC = VC / BC = (BC + VB) / BC

Step 9: Substitute Values and Calculate IC
Substituting the given values into the equation, we can calculate IC.

IC = (BC + VB) / BC = (4V + 4.7V) / 4V = 1.175 A

Step 10: Substitute Values and Calculate β
Substituting the values of IC, IB, and VB into the equation for β, we can calculate the current gain.

β = IC / IB = 1.175 A / 0.0235 mA = 50

Step 11: Answer
Therefore, the current gain is 50.