Test: AC Applied Across Inductor - JEE MCQ

 Answer :- b

Solution :- In a purely inductive circuit (an AC circuit containing inductance only) the current lags behind the voltage by π/2

 Option is not correct because there is relation between current & voltage in inductance

Self-inductance is given by: v(t)=L (di/dt)
From this we have: L=v (dt/di)
 
v/di has the dimensions of R
 
So, L=Rdt hence L/R=dt, same units as that of time.
 

Q-factor: In LCR Circuit, the ratio of resonance frequency to the difference of its neighbouring frequencies so that their corresponding current is 1/√2​ times of the peak value, is called Q-factor of the circuit.
Formula: Q=(1/R)​ √(L/C)​​
So, Q ∝1/R
Conditions for the large value of Q factor:
(i) Value of CL​ should be large.
(ii) Value of R should be less.
 

P(t)=V_mI_mSin²ωt
P(t)=0.5V_mI_m(2sin2ωt)
P(t)= 0.5V_mI_m(1-cos2ωt)
Therefore, frequency is doubled for the instantaneous power so, frequency of instantaneous power is 100Hz.
 

Inductors and chokes are basically coils or loops of wire that are either wound around a hollow tube former (air cored) or wound around some ferromagnetic material (iron cored) to increase their inductive value called inductance.
Inductors store their energy in the form of a magnetic field that is created when a voltage is applied across the terminals of an inductor. The growth of the current flowing through the inductor is not instant but is determined by the inductor's own self-induced or back emf value. Then for an inductor coil, this back emf voltage VL is proportional to the rate of change of the current flowing through it.
This current will continue to rise until it reaches its maximum steady state condition which is around five time constants when this self-induced back emf has decayed to zero. At this point a steady state current is flowing through the coil, no more back emf is induced to oppose the current flow and therefore, the coil acts more like a short circuit allowing maximum current to flow through it.
However, in an alternating current circuit which contains an AC Inductance, the flow of current through an inductor behaves very differently to that of a steady state DC voltage. Now in an AC circuit, the opposition to the current flowing through the coils windings not only depends upon the inductance of the coil but also the frequency of the applied voltage waveform as it varies from its positive to negative values.
 

In a.c. circuits, a choke coil is used to control the current in place of a resistance. If a resistance is used to control the current, the electrical energy will be wasted in the form of heat. Due to its inductive reactance, a choke coil decreases the current without wasting electrical energy in the form of heat.

Cos Ø is called a power factor that indicates what fraction of [(Voltage V) x (Current I)] becomes the useful power. Most electric circuits have resistance and inductance. In such an electric circuit, the current lags the voltage by the phase difference.

Phase Difference between Current and EMF in an Inductance
 

• Explanation: In an inductance, the current lags behind the electromotive force (EMF) by a phase angle of π/2 (90 degrees).


• Reasoning: This phase difference occurs because when the current starts to flow through the inductance, it induces a back EMF that opposes the change in current. This back EMF leads to the current lagging behind the EMF.


• Mathematically: The relationship between the current (I) and the EMF (E) in an inductance can be expressed as: E = L(di/dt), where L is the inductance and di/dt is the rate of change of current with respect to time. This equation shows that the EMF is proportional to the rate of change of current, leading to the phase difference.


 

In pure inductance circuit, R=0
Thus power factor of pure inductance circuit cosϕ=R/Z​=0
Average power dissipation P_av​=V_rms​I_rms​cosϕ
⟹ P_av​=0

Explanation:

- Inductive Circuits:
In purely inductive circuits, the current lags behind the voltage by π/2 (90 degrees). This is because in an inductive circuit, the voltage leads the current due to the phase shift caused by the presence of inductance.

- Phase Relationship:
When an alternating current passes through an inductor, a back EMF is induced in the coil which opposes the flow of current. This causes the current to lag behind the voltage in an inductive circuit.

- Mathematical Representation:
Mathematically, this phase shift can be represented as a phase angle of π/2 (90 degrees) between the current and voltage in a purely inductive circuit.

- Conclusion:
Therefore, in purely inductive circuits, the current lags behind the voltage by π/2 due to the inherent characteristics of inductors to oppose changes in current flow.