Option 2 : Periodically

The correct answer is __ Periodically__.

__Key Points__

__CONCEPT__:

- The
**electric current flows**in two ways: Alternating current and Direct Current.- Direct current flows only in one direction.

- Alternating current: The electric current whose direction changes periodically is called electric current.

- Alternating current reverses its direction periodically.
- It also changes its magnitude periodically because of the induced electromagnetic force.
- For Alternating current both magnitude and direction change. The frequency of the alternating current in the Indian power supply is 50 Hz. The time period is 1/50 = 20 msec.

__ EXPLANATION__:

- In Alternating Current (AC), the
**direction and magnitude of the current vary periodically.**So option 2 is correct.

Option 4 : Increasing voltage

__CONCEPT__:

**Transformer:**

- An
**electrical device**that is used to**transfer electrical energy from one electrical circuit to another**is called a**transformer**.

There are two **types of transformer**:

**1. Step-up transformer**:

- The
**transformer**which**increases the potential**is called a**step-up transformer**. - The
**number of turns in the secondary coil**is**more**than that in the**primary coil**.

**2.Step-down transformer**:

- The
**transformer**which**decreases the potential**is called a**step-down transformer**. - The
**number of turns in the secondary coil**is**less than**that in the**primary coil**.

__EXPLANATION__:

- As in the
**step-up transformer**,**the number of coils in the secondary coil is more than that in the primary coil**. So the**electrical potential increases**. - Hence
**step-up transformer**is**used for increasing the voltage/potential**. So option 4 is correct.

- In
**step-up transformer**, the**current in the secondary coil**is**less than that in the primary coil**.

Option 2 : Minimize eddy current loss

The correct answer is __ Minimize eddy current loss__.

- Transformer cores are laminated in order to minimize core loss.
- By providing laminations, the area of each part gets reduced and hence resistance will get very high which limits the eddy current to a minimum value, and hence
**eddy current losses get reduced.** - The laminations provide small gaps between the plates. As it is easier for magnetic flux to flow through iron than air or coil, the stray flux or leakage flux that can cause core losses is minimized.

Option 4 : 1 / √(LC)

__CONCEPT__:

- The ac circuit containing the capacitor, resistor, and the inductor is called an LCR circuit.
- For a series LCR circuit, the total potential difference of the circuit is given by:

\(V = \sqrt {{V_R^2} + {{\left( {{V_L} - {V_C}} \right)}^2}} \)

Where VR = potential difference across R, VL = potential difference across L and VC = potential difference across C

- For a series LCR circuit, Impedance (Z) of the circuit is given by:

\(\)\(Z = \sqrt {{R^2} + {{\left( {{X_L} - {X_C}} \right)}^2}} \)

Where R = resistance, XL =induvtive reactance and XC = capacitive reactive

__CALCULATION__:

- For a series LCR circuit, Impedance (Z) of the circuit is given by:

\(\)\(\Rightarrow Z = \sqrt {{R^2} + {{\left( {{X_L} - {X_C}} \right)}^2}} \)

- Inductive reactance,

⇒ XL = Lω

- Capacitive reactance

\(\Rightarrow X_c=\frac{1}{C\omega}\)

- Resonance will take place when XL = X
_{C}**.**

****⇒ XL = X_{C}

\(\Rightarrow L\omega =\frac{1}{C\omega}\)

\(\Rightarrow \omega =\frac{1}{\sqrt{LC}}\)

Option 4 : i_{m} sin(ωt + π/2)

**Concept:**

The power factor of the inductor has lagging nature, whereas for capacitor its leading.

For a pure inductor \(\phi = {90^0}\) (The angle between voltage and current)

This shows that if we plot voltage vs current graph the output signal will be lagging by phase angle π/2

The given figure shows the power factor (ϕ) for pure R, L and C circuit

**Explanation:**

From the above explanation, we can see that for a pure capacitance circuit if AC voltage of v = vm sinωt then the driven current in the capacitor will be

I = im sin(ωt + ϕ)

Now as mentioned for pure inductor phase angle will be lagging by 90° or π/2.

i.e., ϕ = π/2 ⇒ i = im sin (ωt + π/2)

Option 2 : Less

__Concept:__

- A transformer is an electrical device which transfers electrical energy from one circuit to another
- It works based on the principle of electromagnetic induction
- It is used for increasing or decreasing the amount of voltage or current as per our requirement based on these transformers are classified into two types step-up (increasing) or step-down (decreasing) transformer.

There are two types of transformer:

1. Step-up transformer:

- The transformer which increases the potential is called a step-up transformer.
- The number of turns in the secondary coil is more than that in the primary coil.

2.Step-down transformer:

- The transformer which decreases the potential is called a step-down transformer.
- The number of turns in the secondary coil is less than that in the primary coil.

__Explanation:__

As in an ideal transformer, there is no loss of power i.e.

Pout = Pin

So, VsIs = VpIp

\(\therefore \frac{{{V_s}}}{{{V_p}}} = \frac{{{N_s}}}{{{N_P}}} = \frac{{{I_P}}}{{{I_S}}}\)

Where Ns = number of turns of the secondary coil, NP = number of turns of the primary coil, Vs = Voltage secondary coil, VP = Voltage primary coil, Is = Current in the secondary coil and IP = current in the primary coil.

**In Step up transformer voltage is increased So, the current will be reduced. As there is no loss in power.**

__Additional Information__

- A transformer can be used to either increase or decrease the voltage of a circuit.
- In other words, it can either step up (increase) or step down (decrease) the voltage.
- A transformer is necessary because sometimes the voltage requirements of different appliances are variable.

Option 4 : ωL

__CONCEPT:__

- The device which stores magnetic energy in a magnetic field is called an inductor.
- A circuit that has the only inductor is known as a purely inductive circuit.

**Inductive reactance**: The resistance offered to the flow of current by an inductor is known as inductive reactance.- It is represented by XL.

- Inductive reactance (X
_{L}) = ω L

Alternating emf in the circuit is:

\(e={{e}_{o}}\sin \omega t\)

Where e0 = maximum potential, ω = angular frequency and t = time

__EXPLANATION__:

**Inductive reactance = ω L**

So option 4 is correct.

Option 3 : leads the voltage by one-fourth of a cycle

__Concept:__

A circuit that has the only capacitor is known as a purely capacitive circuit.

__Explanation:__

When the Alternating emf is running in the circuit is

e = e0 sin ωt

Current in the inductive circuit is –

\(I = {I_0}\sin \left( {\omega t + \frac{\pi }{2}} \right)\)

From above it is clear that current leads the voltage by π/2.

The __Phase difference between current and voltage in the pure capacitive circuit is 90° or π/2.__

A complete cycle makes an angle of 360^{o}, and the Phase difference between current and voltage is 90^{o} **Hence, current leads the voltage by one-fourth of a cycle**

__Important Points__

When the circuit is purely resistive, inductive, and capacitive.

Option 3 : 120 A

__CONCEPT__:

- A Transformer is used to convert low voltage (or high current) to high voltage (or low current) and high voltage to low voltage.
- It works on the principle of electromagnetic induction.
- The primary coil has Np turns and the other coil, called the secondary coil, has Ns turns.
- Generally, the primary coil works as the input coil and the secondary coil works as the output coil of the transformer.

- When an AC voltage is applied to the primary coil, the resulting current produces an alternating magnetic flux that links the secondary coil and induces an emf in it. The value of this emf depends on the number of turns in the secondary.

- In a transformer, the voltage in secondary is calculated by

\(\Rightarrow \frac{N_{s}}{N_{p}}=\frac{V_{s}}{V_{p}}=\frac{i_p}{i_s}\)

Where, Np and Ns are the numbers of turns in the primary and secondary coils respectively, Vp and Vs are the rms voltages across the primary and secondary respectively, ip and is are the current in the primary and secondary coil.

- In a transformer, the load is connected to the secondary coil while the primary coil of a transformer is connected to an AC source.

__EXPLANATION:__

Given - Np = 500, Ns = 200 and ip = 48 A

- The ratio of current in the primary and secondary coil is

\(\Rightarrow \frac{i_p}{i_s}=\frac{N_{s}}{N_{p}}\)

\(\Rightarrow i_s=i_p(\frac{N_p}{N_s})=48\times (\frac{500}{200})=120\, A\)

- Therefore option 3 is correct.

Option 1 : 0

__CONCEPT__:

- Inductor: The device that stores magnetic energy in a magnetic field is called a inductor.

The average power loss (P) in an A.C circuit is given by:

\(P=~{{V}_{rms}}{{I}_{rms}}Cos\theta \)

Where Vrms is RMS voltage in the circuit, Irms is RMS current in the circuit, Φ is the phase angle between the voltage and the current.

__EXPLANATION__:

For a pure inductor circuit:

\(\phi = 90^\circ\) (∵ current lags the voltage by 90° in the pure inductive circuit)

Cos Φ = Cos 90° = 0

\(P = {V_{rms}}{I_{rms}} \times 0\)

\(P = 0\;W\)

- The average power supplied to a inductor over one complete alternating current cycle is 0. So option 1 is correct.

__NOTE__:

- There is no loss of energy in the capacitor and inductor in any circuit. They only store energy.
- The loss of energy is only due to a resistor.

Option 4 : \(M^0L^0T^1\)

__CONCEPT__:

- LC Circuit: The circuit containing both an inductor (L) and a capacitor (C) can oscillate without a source of emf by shifting the energy stored in the circuit between the electric and magnetic fields is called LC circuit.
- The tuned circuit has a very high impedance at its resonant frequency.
- The frequency of oscillations generated by the LC circuit entirely depends on the values of the capacitor and inductor and their resonance condition.
- It can be expressed as:

\(f = \frac{1}{{2π \sqrt {LC} }}\)

Where L = Inductor and C = capacitor

__EXPLANATION__**:**

- Frequency (f): The number of oscillations or the number of waves passes a given point in one sec. Its SI unit is Hertz.

\(\Rightarrow f=\frac{1}{T}\)

- From the above, it is clear that the
**resonant frequency**can be written as

\(\Rightarrow f = \frac{1}{{2π \sqrt {LC} }}\)

The above equation can be written as,

\(\Rightarrow \sqrt{LC}=\frac{1}{2π f}\)

\(\Rightarrow \sqrt{LC}=\frac{T}{2π }\)

Here, 2π is constant, therefore the dimension of \(\sqrt{LC}\) is

\(\Rightarrow \sqrt{LC}=[M^0L^0T^1]\)

Option 1 : lags the voltage by π/2

__CONCEPT:__

- Inductors: The coils of wire that are wound around any ferromagnetic material (iron cored) or wound around a hollow tube that increase their inductive value are called inductors.
- The inductance (L) is measured in Henry (H) and the instantaneous voltage in volts.
- Rate of instantaneous voltage is given by (v = L di/dt)

__EXPLANATION:__

- The given diagram is a simple inductor circuit with alternating current.
- From the phaser diagram, the Inductor current lags inductor voltage by 90° = π/2.

- The plot of current and voltage for this very simple circuit:
- From the current and voltage wave diagram, Inductor current lags inductor voltage by 90°. So option 1 is correct.

- Inductor current lags inductor voltage by 90°.
- Capacitor voltage lags current by 90°.
- In Resister only circuit, Voltage and Current are in the same phase.
- Or we can say there is no lag between current and voltage.

Option 4 : 1.00

__CONCEPT__:

**LCR Circuit**: The ac circuit containing the capacitor, resistor, and the inductor is called an LCR circuit.

For a series LCR circuit, Impedance (Z) of the circuit is given by:

\(\)\(Z = \sqrt {{R^2} + {{\left( {{X_L} - {X_C}} \right)}^2}} \)

Where R = resistance, XL = inductive reactance and XC = capacitive reactive

- Power factor (Cos Φ): The ratio of the true power to the apparent power of an a.c. the circuit is called the power factor.
- Its value varies from 0 to 1.

The power factor (P) of a series LCR-circuit is given by:

\(\cos {\rm{Φ }} = \frac{R}{Z} = \frac{R}{{\sqrt {{R^2} + {{\left( {{X_L} - {X_C}} \right)}^2}} }}\)

Where R = resistance, Z = Impedance, XL = Inductive reactance and XC = Capacitive reactance

__CALCULATION__:

If we have an LCR circuit and it is under resonance:

It means that XL = Xc, hence Z = R and V = VR

\(\text{Impedance }\!\!~\!\!\text{ of }\!\!~\!\!\text{ the }\!\!~\!\!\text{ circuit }\!\!~\!\!\text{ }\left( \text{Z} \right)=~\sqrt{{{R}^{2}}+~{{\left( {{X}_{L}}-{{X}_{C}} \right)}^{2}}}\)

So Z = R

Power factor = Cosϕ = R/Z = R/R = 1

So option 4 is correct.

Option 4 : \(\frac{1}{2\pi\sqrt{LC}}\)

__CONCEPT__:

- The ac circuit containing the capacitor, resistor, and inductor is called an LCR circuit.
- For a series LCR circuit, the total potential difference of the circuit is given by:

\(V = \sqrt {{V_R^2} + {{\left( {{V_L} - {V_C}} \right)}^2}} \)

Where VR = potential difference across R, VL = potential difference across L and VC = potential difference across C

- For a series LCR circuit, Impedance (Z) of the circuit is given by:

\(\)\(Z = \sqrt {{R^2} + {{\left( {{X_L} - {X_C}} \right)}^2}} \)

Where R = resistance, XL =induvtive reactance and XC = capacitive reactive

__CALCULATION__:

- For a series LCR circuit, Impedance (Z) of the circuit is given by:

\(\)\(\Rightarrow Z = \sqrt {{R^2} + {{\left( {{X_L} - {X_C}} \right)}^2}} \)

- Inductive reactance,

⇒ XL = Lω

- Capacitive reactance

\(\Rightarrow X_c=\frac{1}{Cω}\)

- Resonance will take place when XL = XC.

⇒ XL = XC

\(\Rightarrow Lω =\frac{1}{Cω}\)

\(\Rightarrow ω =\frac{1}{\sqrt{LC}}\)

As we know, ω = 2πf

Where f = frequency

\(\Rightarrow f =\frac{1}{2\pi\sqrt{LC}}\)

Option 2 : 1/(ωC)

__CONCEPT:__

- A circuit that has the only capacitor is known as a purely capacitive circuit.

__EXPLANATION:__

**Capacitive reactance**is defined as the resistance offered to the flow of current by the capacitor.- It is denoted by letter X
_{L}.

- It is denoted by letter X
- The
**capacitive reactance**is given by:

\(\Rightarrow{X_c} = \frac{1}{{\omega C}} = \frac{1}{{2\pi \nu C}}\)

Where C = capacitance and f = supply frequency

**Capacitive reactance**is**inversely proportional to the supply frequency**, i.e.

\(\Rightarrow {X_c} \propto \frac{1}{{ f}}\)

- If the
**frequency increases**, the**capacitive reactance decreases**and vice-versa.

- Alternating emf in the circuit is

\(e = {e_o}\sin \omega t\) - Current in the inductive circuit is –

\(I = {I_o}\sin \left( {\omega t + \frac{\pi }{2}} \right)\) - From above it is clear that current leads the voltage by π/2.

Option 3 : i_{m} / √2

__CONCEPT:__

__Root mean square value of Alternating Current and Alternating E.M.F. __-

- The root mean square (r.m.s.) value of a.c. is defined as that value of steady current, which would generate the same amount of heat in a given resistance in a given time, as is done by the a.c., when passed through the same resistance for the same time.
- The r.m.s. value is also called effective value of a.c. or virtual value of a.c. It is represented by Irms or Ieff or Iv.
- The relation between the peak value of a.c. value of current (Io) and r.m.s. value of current is given as –

\({{I}_{rms}}=\frac{{{I}_{o}}}{\sqrt{2}}\)

__EXPLANATION:__

- From above it is clear that the root mean square value of the alternating current is equal to 1/√2 times the peak value.

Option 3 : mutual induction

__CONCEPT__:

- A Transformer is used to convert low voltage (or high current) to high voltage (or low current) and high voltage to low voltage.
**It works on the principle of mutual induction.**- The primary coil has Np turns. The other coil, called the secondary coil, has Ns turns.
- Generally, the primary coil works as the input coil and the secondary coil works as the output coil of the transformer.
- When an AC voltage is applied to the primary coil, the resulting current produces an alternating magnetic flux that links the secondary coil and induces an emf in it. The value of this emf depends on the number of turns in the secondary.

- In a transformer, the voltage in secondary is calculated by \(\frac{N_{s}}{N_{p}}=\frac{V_{s}}{V_{p}}\)

Here,Np and Ns are the numbers of turns in the primary and the secondary coils respectively and Vp and Vs are the rms voltages across the primary and secondary respectively. - In a transformer, the load is connected to the secondary coil while the primary coil of a transformer is connected to an AC source.
**Mutual induction:**By a change in the current in a circuit which is linked to the another by the flux lines of a magnetic field, the production of an electromotive force in the circuit is called mutual induction.

**EXPLANATION:**

- Since in a transformer, there are two coils. voltage in the second coil induces due to flux change in the first coil. This is the basic principle of mutual induction.
**So the transformer works on the principle of mutual induction.**- So the correct answer is
**option 3.**

Option 2 : i_{m}^{2}R/2

__CONCEPT__:

- Power: The rate of work done by an electric current is called power. It is denoted by P. The SI unit of power is the watt (W).

Power dissipation is given by:

Power dissipated, \(P=VI=\frac{{{V}^{2}}}{R}={{I}^{2}}R\)

Where

V = the potential difference across resistance,

I = current flowing

R = resistance.

**Explanation:**

From the above explanation, we can see that power decapitated by any circuit can be expressed as

\(P=VI=\frac{{{V}^{2}}}{R}={{I}^{2}}R\)

Whereas for A.C current the above expression can be expressed as

P = V_{RMS }I_{RMS}

\(P=\frac{{{V}_{m}}}{\sqrt{2}}\times \frac{{{i}_{m}}}{\sqrt{2}}=\frac{{{i}_{m}}{}^{2}R}{2}...(\because {{V}_{m}}={{i}_{m}}R)\)

Hence option 2 is correct among all

Option 3 : 90°

__CONCEPT:__

- A circuit that has the only inductor is known as a purely inductive circuit.

__EXPLANATION:__

- Alternating emf in the circuit is

\(e={{e}_{o}}\sin \omega t\)

- Current in the inductive circuit is

\(I={{I}_{o}}\sin \left( \omega t-\frac{\pi }{2} \right)\)

- From above it is clear that
**voltage leads the current**by**π/2**or**90°**. Therefore option 3 is correct.

Option 2 : 0

__ CONCEPT__:

**Capacitor**: The device that stores electrostatic energy in an electric field is called a capacitor.

The average power loss (P) in an A.C circuit is given by:

\(P=~{{V}_{rms}}{{I}_{rms}}Cos\theta \)

Where Vrms is RMS voltage in the circuit, Irms is RMS current in the circuit, Φ is the phase angle between the voltage and the current.

__ EXPLANATION__:

For a **pure capacitor circuit**:

\(\phi = 90^\circ\) (∵ current leads the voltage by 90° in the pure capacitive circuit)

Cos Φ = Cos 90° = 0

\(P = {V_{rms}}{I_{rms}} \times 0\)

\(P = 0\;W\)

- The
**average power supplied to a capacitor over one complete alternating current cycle is 0**. So option 2 is correct.

__ NOTE__:

- There is no loss of energy in the capacitor and inductor in any circuit. They only store energy.
- The loss of energy is only due to a resistor.