General Physics (calculus based) Class Notes

Dr. Rakesh Kapoor, M.Sc., Ph.D.

Former Faculty-University of Alabama at Birmingham, Birmingham, AL 35294

Electromagnetic Oscillations and Alternating Current


In this chapter we will learn about following topics.

How oscillations are produced in an LC circuit

How adding a resistance to a LC circuit makes the oscillations damped

What is an Alternating current (AC) source.

Learn about a phasor diagram.

How the AC current behaves in single element (R, L, or C) ac circuits.

Analysis of RLC circuit by phasor diagram.

Average power in AC circuits and how to maximize it.

Transmission looses in AC power delivery and how to improve it.


Electromagnetic oscillations in an LC circuit

LC Oscillations

Consider a circuit as shown in figure with a capacitor C and an inductor L.


Suppose there was an initial charge q present on the capacitor. What happen when we short it with an inductor L?

The capacitor will discharge through the inductor and a time varying current i will build up in the inductor L.

If at any instant of time t there is a charge q on the capacitor and current i through the inductor, total energy U in the circuit can be given as


Where Electromagnetic_Oscillations_3.gif and Electromagnetic_Oscillations_4.gif are energy stored in electric and magnetic field respectively.

If there is no loss of energy with time, then according to conservation of energy


Apply following formula to above equation


We get


Now current i through the circuit can also be written as


By substituting the value of i in our change in energy equation we get




Since above equation is true for any value of i, therefore the term in brackets must be zero any time


Now again substitute the value of i=dq/dt, we get


We can rewrite above equation as


Does above equation looks familiar (Mathematically)?

Try to recollect equation describing the motion of a block on a spring (For simple harmonic motion acceleration is proportional to displacement with a negative sign)



Where ω is the angular frequency of the simple harmonic oscillator (spring block system)

Mathematically LC circuit equation and simple harmonic oscillator equation are same.

We can say that the charge in a LC circuit will oscillate with angular frequency ω such that


Where f is the frequency of oscillation. Charge on the capacitor at any time t will be given as


Where φ is the initial phase angle. Similarly current i through the inductor will be given as


Time period of these oscillations will be given as


Time variation of energy Electromagnetic_Oscillations_20.gif and Electromagnetic_Oscillations_21.gif

Let us see how energy Electromagnetic_Oscillations_22.gif and Electromagnetic_Oscillations_23.gif change in this circuit.



By plugging the value of Electromagnetic_Oscillations_26.gif in above equation we get


A plot of total energy U, Electromagnetic_Oscillations_28.gif and Electromagnetic_Oscillations_29.gif is shown below.


You can see that at T/4 ,3T/4 time, electric field energy goes to zero (Minimum) and current goes to its maximum value.

This is only true for an isolated LC circuit. In practical situation any LC circuit can get coupled with outside world and start transferring its energy to other inductors in the neighbor hood.

Checkpoint 1

A capacitor in an isolated LC oscillator has a maximum potential difference of 17 V and a maximum energy of 160 μJ. When the capacitor has a potential difference of 5 V and an energy of 10 μJ, what are (a) the emf across the inductor and (b) the energy stored in the magnetic field?


Total energy is conserved. The potential difference across L and C should always be same.

Damped Oscillations in RLC Circuit:

If we add a resistor to the LC circuit as shown in figure.


When ever the current will flow through the circuit, some energy will be dissipated in the resistor. The rate of dissipation of energy is given as


Now the total rate of change of energy equation can be rewritten as




By dividing both sides with i and substituting the value of i=dq/dt, we get



If you look it carefully, mathematically this is similar to the damped harmonic oscillator equation


This equation has the solution




Similarly for damped oscillation of charge in a LRC circuit can be given as




This equation describes oscillation of charge or current in the circuit with decaying amplitude Electromagnetic_Oscillations_42.gif.

Damped angular frequency ω' is always smaller than the undamped angular frequency ω.


Quiz- 1
Which one of the following quantities remains constant for a given isolated LC circuit?

the energy stored in the capacitor

the energy stored in the inductor

the energy stored in the current flowing in the circuit

the sum of the energy stored in the capacitor and that in the inductor

the energy dissipated in the circuit

Quiz- 2
When the current in an oscillating LC circuit is zero.  Which one of the following statements is true?

The charge on the capacitor is equal to zero coulombs.

Charge is moving through the inductor.

The energy is equally shared between the electric and magnetic fields.

The energy in the electric field is maximized.

The energy in the magnetic field is maximized.

Quiz- 3
Which one of the following is the correct expression for the angular frequency of oscillation for an LC circuit?

ω = LC

ω = √LC

ω = 1/√LC

ω = 1/(LC)

ω = 2π/√LC

Quiz- 4
Which one of the following choices will damp oscillations in an RLC circuit?

increase the inductance

increase the emf

increase in both inductance and capacitance

increase the capacitance

increase the circuit resistance

Alternating current (AC) circuits

Direct Current (dc.)

A battery emf E  always generates a current that has a constant direction. This type of current is known as "direct current or dc".

Alternating Current (ac.)

We are familiar with the device shown in following figure.


This is a different type of emf generating source. The emf E produced by such a source is given as


Where A is the area of the generator loop, N is the number of loops or windings, ω is the angular frequency of rotation and B is the magnetic field.

In this type of device (generator) emf E change direction with a frequency 2π f=ω. In US f=60 Hz.

Because the emf E change direction, therefore current i through the connected circuit will also change direction with frequency f. Such type of current is known as "alternating current or ac."

Symbol for an alternating current source is shown in the following figure.


Symbol convention :

To analyze ac circuits, we will use the following symbol convention.

Lower case letter will be used to indicate instantaneous value of an ac quantity (like current i etc.)

Upper case letters will be used to indicate constant amplitude of ac quantities.

Example, ac current i will be written as


Symbol i is for instantaneous value of current and I is its maximum value.


A convenient method of representing ac quantities like Electromagnetic_Oscillations_53.gif and Electromagnetic_Oscillations_54.gif is a phasor diagram.

In phasor diagram, the ac quantities like Electromagnetic_Oscillations_55.gif and Electromagnetic_Oscillations_56.gif are represented by a rotating vector known as phasor using the following conventions.


Phasor rotate in counterclockwise direction with angular speed ω.

The length of phasor is proportional to the amplitude of the ac quantity.

Projection of the phasor on vertical axis (y-component) gives the instantaneous value of the ac quantity.

Rotation angle ω t for each phasor is called the phase of the quantity.

Three Simple Circuits

Our objective is to analyze a RLC circuit connected to an ac emf source.


Analysis of an RLC circuit becomes simple and easy if we can analyze three individual elements (R, C, and L) connected to an ac emf source.

A Resistive load :

Let us consider a resistance R connected to an ac generator.


According to Kirchhoff s law we have


The current Electromagnetic_Oscillations_61.gif through the resistance will be given as


The amplitude of Electromagnetic_Oscillations_63.gif is given as


The voltage Electromagnetic_Oscillations_65.gif across R is


The relation between voltage and current amplitude is


Variation of Electromagnetic_Oscillations_68.gif and Electromagnetic_Oscillations_69.gif as a function of time (Resistive load)

It can be seen that both Electromagnetic_Oscillations_70.gif and Electromagnetic_Oscillations_71.gif reach to their respective maximum and minimum values at the same time.


For a resistive load Electromagnetic_Oscillations_73.gif and Electromagnetic_Oscillations_74.gif are always in phase.

Phasor diagram of a resistive load


In this phasor diagram the phase of Electromagnetic_Oscillations_76.gif and Electromagnetic_Oscillations_77.gif is same at any instant.

Phasor Electromagnetic_Oscillations_78.gif and Electromagnetic_Oscillations_79.gif always points in the same direction at any instant of time.

By convention Electromagnetic_Oscillations_80.gif is written as


Voltage Electromagnetic_Oscillations_82.gif is written as


For a resistor, the initial phase angle Electromagnetic_Oscillations_84.gif.

Average Power in a resistive load

As we have seen that in an ac circuit, the current Electromagnetic_Oscillations_85.gif and voltage Electromagnetic_Oscillations_86.gif across R are changing with time, therefore dissipated power P(t) is a function of time.

To estimate dissipated power in the resistor we have to calculate average power <P> in one cycle or in one time period T.


By substituting the values of Electromagnetic_Oscillations_88.gif and Electromagnetic_Oscillations_89.gif we get.




Therefore average power dissipated in a resistive load R in an ac circuit is given as


To make the equation look same as in the dc case, a new quantity called "root mean square" voltage Electromagnetic_Oscillations_93.gif is defined


Now the dissipated average power <P> in a resistive load can be written as


Similarly "root mean square" current is defined as Electromagnetic_Oscillations_96.gif


Checkpoint 2

If we increase the driving frequency in a circuit with a purely resistive load, do (a) amplitude Electromagnetic_Oscillations_98.gif and (b) amplitude Electromagnetic_Oscillations_99.gif increase, decrease, or remain the same?

Hint: Amplitude Electromagnetic_Oscillations_100.gif is related to amplitude Electromagnetic_Oscillations_101.gif and Electromagnetic_Oscillations_102.gif.

Quiz- 5
The graph shows the current as a function of time for an electrical device plugged into a outlet with an rms voltage of 120 V.  What is the resistance of the device?

24 Ω

21 Ω

17 Ω

14 Ω

12 Ω

A Capacitive Load

Connect an ac emf source to a capacitor C.


If at a given instant Electromagnetic_Oscillations_105.gif is the potential difference across the capacitor, then according to Kirchhoff s law


The charge Electromagnetic_Oscillations_107.gif on the capacitor at any instant is given as


The current Electromagnetic_Oscillations_109.gif through the circuit at any instant is given as


Above equation can also be written as


Amplitude of voltage Electromagnetic_Oscillations_112.gif across the capacitor is equal to amplitude of applied emf Electromagnetic_Oscillations_113.gif. Electromagnetic_Oscillations_114.gif, therefore the current amplitude is


This convention is followed to make this equation look similar to resistive load. The quantity Electromagnetic_Oscillations_116.gif is called "Capacitive reactance or Capacitive impedance". The current through such a circuit can be written as


You can see that current Electromagnetic_Oscillations_118.gif through a capacitive load is Electromagnetic_Oscillations_119.gif out of phase with respect to voltage across the capacitor.

If we write Electromagnetic_Oscillations_120.gif as


The phase angle Electromagnetic_Oscillations_122.gif for a capacitor.

In complex number notation "capacitive reactance" is written as


Here Electromagnetic_Oscillations_124.gif is "iota". In complex number notation Electromagnetic_Oscillations_125.gif can also be written as

Capacitive impedance decreases with increase in capacitance and angular frequency ω


Current in a capacitor leads the voltage by π/2 phase.


In terms of phasor diagram, the current and voltage across a capacitor connected to an ac source, at any instant is shown below.


Average Power in a capacitive load

Power P(t) dissipated in the capacitor at any instant is given as


By using the trigonometric relation


we get


Average dissipated power <P> in the capacitor can be calculated as below


This shows that a capacitor does not dissipate any power on the average.

In some part of the cycle it absorbs the power and store it as electric field energy, in the other part of the cycle it gives it back to the circuit.

In the end on the average no power is used by a capacitive load in an ac circuit.

Checkpoint 3

The figure shows, in (a), a sine curve Electromagnetic_Oscillations_133.gif and three other sinusoidal curves A(t), B(t), and C(t), each of the form Electromagnetic_Oscillations_134.gif. (a) Rank the three other curves according to the value of φ, most positive first and most negative last. (b) Which curve corresponds to which phasor in figure-b? (c) Which curve leads the others?


Hint: Remember if one sin curve looks ahead of the other sin curve, it means it is lagging behind the other. In phasor diagram net positive phase goes in counter clockwise direction.

Quiz- 6
Five identical capacitors are connected in three different circuits as shown.  The ac current source in each circuit is the same.  In which circuit is the rms current the largest and in which is the rms current the smallest?

C has the largest rms current and A has the smallest.

A has the largest rms current and C has the smallest.

B has the largest rms current and A has the smallest.

B has the largest rms current and C has the smallest.

C has the largest rms current and B has the smallest.

An Inductive Load

Connect an ac emf source to an Inductor L.


The voltage Electromagnetic_Oscillations_138.gif across the inductor L at any instant is related to change in current Electromagnetic_Oscillations_139.gif through the circuit.


According to Kirchhoff s law we have





Above equation can also be written as


Amplitude of voltage Electromagnetic_Oscillations_145.gif across the inductor is Electromagnetic_Oscillations_146.gif.

The current amplitude is


The quantity Electromagnetic_Oscillations_148.gif is called "Inductive reactance or Inductive impedance".

You can see that current Electromagnetic_Oscillations_149.gif through an inductive load is Electromagnetic_Oscillations_150.gif out of phase with respect to voltage across the inductor.

In other words the current in an inductor lags behind the voltage by π/2 phase.

If we write Electromagnetic_Oscillations_151.gif as


The phase angle Electromagnetic_Oscillations_153.gif for an inductor.

In complex number notation "Inductive reactance" is written as


Inductive impedance Increases with increase in inductance and angular frequency ω

In complex number notation Electromagnetic_Oscillations_155.gif is written as


-i means -π/2 phase difference.

Current in an inductor lags behind the voltage by π/2 phase.


In terms of phasor diagram, the current and voltage across an inductor connected to an ac source, at any instant is shown below.


Average Power in an inductive load

Power P(t) dissipated in the inductor at any instant is given as


Again by using the trigonometric relation


we get


Average dissipated power <P> in the inductor can be calculated as below


This shows that an inductor does not dissipate any power on the average.

In some part of the cycle it absorbs the power and store it as magnetic field energy, in the other part of the cycle it gives it back to the circuit.

In the end on the average no power is used by an inductive load in an ac circuit.

Checkpoint 4

If we increase the driving frequency in a circuit with a purely capacitive load, do (a) amplitude Electromagnetic_Oscillations_163.gif and (b) amplitude Electromagnetic_Oscillations_164.gif increase, decrease, or remain the same? If, instead, the circuit has a purely inductive load, do (c) amplitude Electromagnetic_Oscillations_165.gif and (d) amplitude Electromagnetic_Oscillations_166.gif increase, decrease, or remain the same?

Hint: Voltage across the terminals of the load will always be equal to voltage across the emf source.  Capacitive reactance Electromagnetic_Oscillations_167.gif. Inductive reactance is Electromagnetic_Oscillations_168.gif. Magnitude of current I=V/X.

Quiz- 7
An inductor circuit operates at a frequency f = 120 Hz.  The peak voltage is 120 V; and the peak current through the inductor is 2.0 A.  What is the inductance of the inductor in the circuit?

0.040 H

0.080 H

0.16 H

0.32 H

0.64 H

Resonance in RLC circuits

Connect an ac emf source E to a RLC circuit.



Since R, L and C are in series, same current i should flow through all the elements. Let us assume that the current through the circuit is


If we draw the phasor vector for current I at any instant of time, we can draw the phasor vectors of Electromagnetic_Oscillations_172.gif for resistor R, Electromagnetic_Oscillations_173.gif for inductor L and  Electromagnetic_Oscillations_174.gif for capacitor C.


Electromagnetic_Oscillations_176.gif is in phase with the current phasor vector.

Electromagnetic_Oscillations_177.gif is Electromagnetic_Oscillations_178.gif ahead of the current phasor vector.

Electromagnetic_Oscillations_179.gif is Electromagnetic_Oscillations_180.gif behind the current phasor vector.

From the loop rule the emf E at any time t is the sum of voltages Electromagnetic_Oscillations_181.gif, Electromagnetic_Oscillations_182.gif and Electromagnetic_Oscillations_183.gif.


In a loop, at any time current Electromagnetic_Oscillations_185.gif through all the loads is same, therefore


By substituting value of Electromagnetic_Oscillations_187.gif and Electromagnetic_Oscillations_188.gif we get.


Here Z is the complex impedance  of the RLC circuit.


Magnitude and phase of impedance can be given as



Current through such a RLC circuit will have φ phase difference with applied emf.


On phaser diagram, emf E is the sin component (projection on vertical axis) of the phasor vector Electromagnetic_Oscillations_194.gif. when sin component of a phasor is the sum of sin components Electromagnetic_Oscillations_195.gif, Electromagnetic_Oscillations_196.gif and Electromagnetic_Oscillations_197.gif of other phasors. That implies that the phasor  Electromagnetic_Oscillations_198.gif is the phasor (vector) sum of three phasors Electromagnetic_Oscillations_199.gif, Electromagnetic_Oscillations_200.gif and Electromagnetic_Oscillations_201.gif.

Since Electromagnetic_Oscillations_202.gif and Electromagnetic_Oscillations_203.gif are opposite to each other, we can combine these two phasors as Electromagnetic_Oscillations_204.gif. Now Electromagnetic_Oscillations_205.gif is the phasor sum of Electromagnetic_Oscillations_206.gif and Electromagnetic_Oscillations_207.gif.


By applying Pythagoras theorem to above phasor diagram, we get


In terms of current I and reactance Electromagnetic_Oscillations_210.gif, Electromagnetic_Oscillations_211.gif and resistance R, we can rewrite the above equation as


From above equation we get the value of current amplitude I.


Denominator is called the impedance Z of the circuit for the driving angular frequency Electromagnetic_Oscillations_214.gif.


We can write





Similarly above diagram can be used to calculate the phase constant


There can be three possibilities :

Electromagnetic_Oscillations_220.gif, implies phase angle φ>0. Current phasor lags behind the emf phasor. Circuit is more inductive.  

If Electromagnetic_Oscillations_221.gif, implies phase angle φ<0. Current phasor leads the emf phasor. Circuit is more capacitive.

If Electromagnetic_Oscillations_222.gif, implies phase angle φ=0. Current phasor and emf phasor are in phase.


Let us look at the current amplitude equation.


For a given value of R, the current amplitude will be maximum when




We can say that the current through a RLC circuit for a given value of R will be maximum when the natural angular frequency Electromagnetic_Oscillations_227.gif is equal to the driving angular frequency of the emf source.


This condition is called resonance condition.

In other words we can state that in a RLC circuit when inductive reactance Electromagnetic_Oscillations_229.gif is equal to the capacitive reactance, the circuit is in resonance condition and maximum current flows through the circuit.

How can we achieve resonance?

By changing the inductance L of the circuit such that Electromagnetic_Oscillations_230.gif.

By changing the capacitance C of the circuit such that Electromagnetic_Oscillations_231.gif.

By changing the driving angular frequency Electromagnetic_Oscillations_232.gif of emf source such that Electromagnetic_Oscillations_233.gif.

Variation of current amplitude in a RLC circuit.

Variation of current phase in a RLC circuit.

Checkpoint 4

Here are the capacitive reactance and inductive reactance, respectively, for three sinusoidally driven series RLC circuits: (1) 50 Ω, 100 Ω; (2)  50 Ω, 50 Ω;(3)  100 Ω, 50 Ω. (a) For each, does the current lead or lag the applied emf, or are the two in phase? (b) Which circuit is in resonance?

Hint: If Electromagnetic_Oscillations_236.gif, φ is negative. If Electromagnetic_Oscillations_237.gif, φ is positive. Resonance condition is Electromagnetic_Oscillations_238.gif.

Quiz- 8
A circuit contains a AC voltage source, a resistor, and another component.  The voltage amplitude is held constant and the frequency is increased.  The current in the circuit is observed to increase as the frequency is increased.  What is the additional component in the circuit?




light bulb

Power in AC circuits

Let us now calculate average power dissipated or used by a RLC circuit. We know that average power is dissipated only in the resistance R of the circuit. Instantaneous power P(t) dissipated in a R of circuit is given as


Average of Electromagnetic_Oscillations_240.gif is 1/2, therefore average power <P> dissipated in a RLC circuit is


We can write it in terms of root-mean-square, or rms current Electromagnetic_Oscillations_242.gif term


In a RLC circuit, Electromagnetic_Oscillations_244.gif is related to rms value of emf amplitude Electromagnetic_Oscillations_245.gif and impedance Z.


Now average power  <P> dissipated is given as


If we look at the phase diagram of RLC circuit, the ratio R/Z is equal to the cosine of phase φ.



  <P> is given as


Term Cos(φ) is called power factor.

To maximize the transmitted power in a RLC circuit, value of Cos(φ) should be one, or the phase angle φ should be as close to zero as possible.

If a circuit is highly inductive, it can be made less by reducing the capacitance value in the circuit or putting a capacitance in series as that will reduce the equivalent capacitance in the circuit.

To achieve maximum power transmission, power companies place series connected capacitors throughout their transmission systems.

Checkpoint 5

(a) If the current in a sinusoidally driven series RLC circuit leads the emf, would we increase or decrease the capacitance to increase the rate at which energy is supplied to the resistance? (b) Would this change bring the resonant angular frequency of the circuit closer to the angular frequency of the emf or put it farther away?

Hint: If current leads the emf, φ is negative, implies Electromagnetic_Oscillations_251.gif. It means we need to reduce the value of Electromagnetic_Oscillations_252.gif Resonance condition is Electromagnetic_Oscillations_253.gif.

Transformers, AC power transmission

Energy Transmission Requirements

Average power delivered by a power station of rms emf Electromagnetic_Oscillations_254.gif with power factor Cos(φ)=1, is


The power dissipated Electromagnetic_Oscillations_256.gif in the transmission lines of resistance R will be given as


Power delivered by a power station is constant, it can be seen that the rms current Electromagnetic_Oscillations_258.gif, will increase if we reduce the rms emf Electromagnetic_Oscillations_259.gif.

The increase in Electromagnetic_Oscillations_260.gif will cause increase in transmission power loss Electromagnetic_Oscillations_261.gif.

Problem 1

Typical resistance for a transmission line is 0.220Ω/km. Power station delivered 350 MW power on a 1000 km transmission line. (a) If the rms emf Electromagnetic_Oscillations_262.gif, what percentage of total power is lost in the transmission line? (b) If we reduce the rms emf to 375 kV, what percentage of power will be lost in the transmission line?

Solution :

For a 1000 km transmission line R=220Ω.

(a) When Electromagnetic_Oscillations_263.gif, Electromagnetic_Oscillations_264.gif will be given as

Electromagnetic_Oscillations_265.gif, Electromagnetic_Oscillations_266.gif in Amperes is



Electromagnetic_Oscillations_269.gif in Watts is



Percentage loss is



(b) When Electromagnetic_Oscillations_274.gif, Electromagnetic_Oscillations_275.gif in Amperes is



Electromagnetic_Oscillations_278.gif in Watts is



Percentage loss is



You can see just by doubling the Electromagnetic_Oscillations_283.gif, the transmission losses increases from 14% to 55%.

The Ideal Transformer:

Reduction in transmission loss requires high voltage.

Safe operations at home require low voltage.

To achieve this we use a device called Transformer.

Transformer is a device that can change the voltage amplitude of an ac emf source.

It consists of two coils wound on an iron core. The coil on which ac emf source is connected is called "primary core".

The core which delivers emf to the circuit is called "secondary coil".


If Electromagnetic_Oscillations_285.gif is the emf amplitude on the primary side and Electromagnetic_Oscillations_286.gif is the emf amplitude on the secondary side, in an ideal transformer, the ratio of two is related to the number of turns Electromagnetic_Oscillations_287.gif of primary coil and number of turns Electromagnetic_Oscillations_288.gif of secondary coil.




When Electromagnetic_Oscillations_291.gif the transformer is called "step down transformer". In this transformer Electromagnetic_Oscillations_292.gif.

When Electromagnetic_Oscillations_293.gif the transformer is called  "step up transformer". In this transformer Electromagnetic_Oscillations_294.gif.

Current in primary coils :

Current through secondary coil induce an emf in primary which affects the current in primary coil.

Estimation of current in primary coil by taking all kind of inductions into account can be involved but by conservation of energy argument, we can find it more easily.

If R is the load in secondary loop, the power Electromagnetic_Oscillations_295.gif dissipated in secondary loop should be


If Electromagnetic_Oscillations_297.gif is the current in primary coil, power Electromagnetic_Oscillations_298.gif delivered by primary circuit is


As per conservation of energy, the power dissipated in secondary loop should be equal to the power Electromagnetic_Oscillations_300.gif delivered by primary circuit.




By rewriting above equation we get the value of primary loop current Electromagnetic_Oscillations_303.gif.


When R is the load in secondary circuit of a transformer, It is equivalent to say that primary coil is connected to a load Electromagnetic_Oscillations_305.gif. Where equivalent load Electromagnetic_Oscillations_306.gif is given as


Impedance Matching :

Any emf source (dc or ac) delivers maximum power to a load, when the value of load is equal to the internal resistance (for a dc source) or internal impedance (for an ac source).

Checkpoint 6

An alternating-current emf device in a certain circuit has a smaller resistance than that of the resistive load in the circuit; to increase the transfer of energy from the device to the load, a transformer will be connected between the two. (a) Should Electromagnetic_Oscillations_308.gif be greater than or less than Electromagnetic_Oscillations_309.gif? (b) Will that make it a step-up or step-down transformer?

Hint: For maximum delivery of energy, equivalent resistance Electromagnetic_Oscillations_310.gif should be equal to the internal resistance of the emf source.