Physics 2004 JAMB Past Questions - Question 29: ac circuit of emf 12v has a resistor of resistance 8 connected in series to an inductor of inductive reactance 16 and a capacitor of capacitive reactance 10 the current flow in the circuit is

Correct Answer

B

Explanation

To find the current flowing in the AC circuit, you can use the impedance concept in AC circuits. Impedance is the total opposition to the flow of alternating current and depends on the components in the circuit.

In this circuit, you have a resistor (R), an inductor (L), and a capacitor (C) connected in series. The impedance (Z) for each component is given by:

1. Impedance of the resistor (Z_R) = R
2. Impedance of the inductor (Z_L) = jωL, where j is the imaginary unit, ω is the angular frequency, and L is the inductance.
3. Impedance of the capacitor (Z_C) = -j/ωC, where j is the imaginary unit, ω is the angular frequency, and C is the capacitance.

The angular frequency (ω) is related to the frequency (f) of the AC signal by the formula ω = 2πf.

Given that the inductive reactance (X_L) is 16 Ω and the capacitive reactance (X_C) is 10 Ω, you can relate these to the impedance of the inductor and the capacitor as follows:

X_L = ωL => 16 = ωL
X_C = -1/(ωC) => 10 = -1/(ωC)

Now, you can calculate ω:

ω = 16/L => ω = 16/ωC

Now, solve for ω:

ω = √(160) = 4√10

With ω known, you can calculate the impedance of the inductor and capacitor:

Z_L = jωL = j(4√10)(16) = j64√10
Z_C = -j/(ωC) = -j/(4√10)(10) = -j√10

The total impedance in a series circuit is the sum of the impedances of the individual components:

Z_total = Z_R + Z_L + Z_C
Z_total = 8 + j64√10 - j√10

Now, we have the total impedance. To find the current (I) flowing in the circuit, you can use Ohm's law for AC circuits:

I = EMF / Z_total

Given that the electromotive force (EMF) is 12 V, you can calculate the current:

I = 12 V / (8 + j64√10 - j√10)

To calculate this complex number, you can rationalize the denominator by multiplying both the numerator and denominator by the conjugate of the denominator:

I = (12 V / (8 + j64√10 - j√10)) * ((8 - j64√10 + j√10) / (8 - j64√10 + j√10))

Now, multiply out the terms:

I = (12 V * (8 - j64√10 + j√10)) / ((8 + j64√10 - j√10) * (8 - j64√10 + j√10))

I is a complex number, and you can calculate its magnitude (absolute value) and phase (angle) to find the current magnitude and phase angle in the circuit. The magnitude will give you the current magnitude, and the phase angle will give you the phase relationship between the current and voltage.

Keep in mind that I may be a complex number, and the exact calculations will depend on the specific values of the constants and the frequency of the AC signal.