Rank the magnitude of current induced into a loop by a time-dependent current

in a straight wire.

I(t) A B C D E

t I(t)Iinduced

What is the direction of the induced current in the loop?

• Zero

• counterclockwise

• clockwise

• Some other

direction

What is the direction of the induced current in the loop?

• Zero

• counterclockwise

• clockwise

• Some other

direction

S N

When the switch is closed, the potential difference across R is…

V N1 N2>N1

• Zero R

• V N2/N1

• V N1/N2

• V

Once the switch is closed, the ammeter shows…

V N1 N2>N1

A

R• Zero Current• Steady Current• Nonzero Current for a short time

A transformer is fed the voltage signal Vp(t).

What is the secondary voltage signal? Vp(t)

t

The concept of self-inductance resembles …

• Aging; the older you get, the weaker you get

• Being taxed; the more money you make, the more taxes you pay

• Swimming; the more you press the water away, the harder the water presses back

• Baron Muenchausen; he pulled himself out of the swamps by his own hair

Self-inductance L is analogous to …

• Electric charge

• Potential energy

• (Inertial) mass

• Momentum

(Hint: it has to be a property of an object)

Given are the potential energies U as a function of the current I of several

inductors. Which has the smallest self-induction L?

U

I

U

U

U

I I

I

A Solenoid produces a changing magnetic field that induces an emf which lights bulbs

A & B. After a short is inserted, …• A goes out, B brighter

• B goes out, A brighter

• A goes out, B dimmer

• B goes out, A dimmer

x x x

A x x B

A straight wire carries a constant current I. The rectangular loop is pushed towards the straight

wire. The induced current in the loop is …

• Zero

• Clockwise I

• Counter-clockwise

• Need more information

When the switch is closed in an LR circuit, the current exponentially reaches the

maximal value I=V/R. The time constant is τ =L/R. After which time does the current

reach half its maximal value?• Immediately• After 1 time constant (t = τ)

• After 2 time constants (t = 2τ)

• After about 70% of a time constant (t = 0.69τ )

When the switch is closed in an LR circuit, the current exponentially reaches the maximal value

I=V/R. If the inductor (a solenoid, say), is replaced by a solenoid with twice the number of windings, the time the current takes to reach half

its maximal value …?• does not change

• is halved

• doubles

• quadruples

The current in an LC circuit will oscillate with a frequency f. To change the frequency we can…

• … change the initial charge of the capacitor

• … change the inductance of the inductor

• …do nothing. It is fixed by the physics of LC circuits.

The current in an LC circuit will oscillate with a frequency f. If we replace the capacitor by one

with twice its capacitance, the frequency …

• doubles

• quadruples

• is halved

• None of the above.

The current in an LC circuit will oscillate with a frequency f. If we add a small

resistance to the circuit, …

• the frequency goes up

• the amplitude goes down

• the current decays exponentially

• Two of the above

Assume a sinusoidal current: I=I0sinω t. In an AC circuit with a resistor R, which

diagram describes the voltage across the resistor correctly?

V

t

V

V

V

t t

t

Assume a sinusoidal current: I=I0sinω t. In an AC circuit with an inductor L, which diagram describes the voltage across the

inductor correctly?

V

t

V

V

V

t t

t

Assume a sinusoidal current: I=I0sinω t. In an AC circuit with an capacitor C, which diagram describes the voltage across the

capacitor correctly?

V

t

V

V

V

t t

t

Resistor

Inductor

• Potential difference (voltage) gets current flowing

• Induction slows current down

Voltage first!

Capacitor

• Flow of charges (current) builds up electric field (voltage)

Current first!

LRC circuit with AC driving emf

• Voltages different: VR , VL , VC

• Common to all: current

Use current as reference

Phasors

• Phasors turn with angular frequency ω

• Direction is position within cycle

• Length of phasor is peak value of V, I, Z

• Value is projection on y axis

• E.g.: VC=0, VR=VR0

A little later …

• ALL phasors

have turned by an angle ωt

• Angles between phasors are preserved

• ALL values of V, I have changed

• E.g.: VL(t=later) = VL0sin (ωt+π/2)

Projections on x-axis are values at time t

Adding Phasors

• Add like vectors

• Phase angle will be between 90 and -90

Assume a sinusoidal current: I=I0sin(2πf t). In a resistor circuit with frequency 2 Hz, which

phasor diagram describes the voltage across the resistor at t = 1.5 s if the phase at t=0 was zero?

Assume a sinusoidal current: I=I0sinω t. In an AC circuit with a capacitor C, which

phasor diagram describes the voltage across the capacitor correctly?

Assume a sinusoidal current: I=I0sinω t. In an AC circuit with a inductor I, which

phasor diagram describes the voltage across the inductor correctly?

Assume a sinusoidal current: I=I0sin(2πf t). What can you tell from the phasor diagram below about an LRC ac circuit if the orange arrow represents the instantaneous voltage

across the whole circuit?The frequency is 1/8 HzThe phase angle of the current is about 30 degreesThe inductive reactance is smallerthan the capacitive reactanceThe resistance is very small

Assume a sinusoidal current: I=I0sin(2πf t). Which of the following is true about an LRC ac

circuit?

The phase angle between current and voltage constantThe voltage is constantThe power consumed by the circuit is zeroThe power consumed by the circuit is constant

Group Work on AC LRC circuits

• L=200mH, R=1000 Ohm, C = 60μF, driven by a 30V power supply at 1kHz.

• Draw the voltages and the current in a phasor diagram at t=1/4000 s.

• Calculate the reactances• Calculate the impedance of the circuit• Find the phase angle • What is the (average) power used by the circuit?

In Physlet I 31.7 an RC circuit is animated. What happens if the frequency increases?

Nothing except the voltage phasor rotating fasterThe reactance of the capacitor goes up and hence the phase angle between voltage and current changes All reactances (R, C) changeThe reactance of the inductor goes up, of the capacitor goes down, and the voltage phasor rotates faster

In Physlet I 31.7 an RC circuit is animated. What will happen if the frequency is

halved?

The reactance of the resistor halvesThe reactance of the capacitor doubles The peak voltage across the source changesThe phase angle between the voltage across R and the voltage across C changes

In Physlet I 31.7 an RC circuit is animated. What will happen if the frequency is

halved?

The voltage across the resistor halvesThe voltage across the capacitor doubles The peak voltage across the source changesNone of the above

Why does the voltage across the capacitor not double if the frequency is halved?

The reactance of the capacitor does not doubleThe current through the circuit dropsThe peak voltage across the source does not changeThe phase angle between the voltages does not change enough

Consider a LRC circuit which at f = 1kHz displays R=XC=XL=1000Ω.

At 10 kHz we have …

• R=XC=XL=1000Ω

• R=1000Ω, XC > XL=10000Ω

• R = XC = XL=10000Ω

• R=1000Ω, XC =100Ω < XL

Consider a LRC circuit which at f = 1kHz displays R=XC=XL=1000Ω.

At 10 Hz we have …

• R=XC=XL=1Ω

• R=1000Ω, XC > XL=1Ω

• R = XC = XL=1MΩ

• R=1000Ω, XC =100Ω < XL

In Physlet I 31.8 the impedance Z of a LRC circuit is plotted. What

happens if the value for R is chosen very big?

• Plot changes, but remains qualitatively the same

• Nothing changes

• The curve is shifted up

• The curve becomes flat

The impedance of a LRC circuit depends on the frequency. What is special about the frequency where capacitor and the inductor have the

same reactance?• Nothing

• Impedance has a minimum

• Current is at a minimum

• All voltages are in phase

As one of Maxwell’s equations, Gauss’s Law is …

• Homogeneous and concerning the electric field

• Inhomogeneous and concerning the electric field

• Homogeneous and concerning the magnetic field

• Inhomogeneous and concerning the magnetic field

As one of Maxwell’s equations, (modified) Ampere’s Law is …

• Homogeneous and concerning the electric field

• Inhomogeneous and concerning the electric field

• Homogeneous and concerning the magnetic field

• Inhomogeneous and concerning the magnetic field

As one of Maxwell’s equations, magnetic Gauss’s Law is …

• Homogeneous and concerning the electric field

• Inhomogeneous and concerning the electric field

• Homogeneous and concerning the magnetic field

• Inhomogeneous and concerning the magnetic field

As one of Maxwell’s equations, Faraday’s Law is …

• Homogeneous and concerning the electric field

• Inhomogeneous and concerning the electric field

• Homogeneous and concerning the magnetic field

• Inhomogeneous and concerning the magnetic field

Electromagnetic Waves

• Medium = electric and magnetic field• Speed = 3 105 km/sec

Production of EM waves

• Current flowing creates B field

• Charges accumulating create E field

EM Waves radiating out

• As the direction of the current changes, the “second half” of the wave is created

E, B in opposite direction as in first half, but in same direction as in “back part” of first half

Wave travels in empty space

Directions of E, B are perpendicular but in phase

• E, B are perpendicular to direction of motion of wave transverse wave

The EM spectrum

Receiving an EM Wave