RI deliveredpower Average d)(

I2 current Maximum c)(

RI 2or V2V gePeak volta b)(

RI V :Law sOhm' a)(

2

rns

rms

rmsrmsmax

rmsrms

=

=

==

=

HW#14 Last homework set

1. When an AC source is connected across a resistor R, the rms current in the resistor is Irms.

(a) Find the rms voltage across the resistor. (b) Find the peak voltage of the source.

(c) Find the maximum current in the resistor. (d) Find the average power delivered to the

resistor.

R.greater for the less is W V, same for the .R

V W Since c)(

W

VR W

R

V b)(

V2V2V gePeak volta a)(

2

22

rmsmax

∴=

=⇒=

===

2. A certain lightbulb is rated at W when operating at an rms voltage of V.

(a) What is the peak voltage applied across the bulb? (b) What is the resistance of the bulb? (c)

Does a 100-W bulb have greater or less resistance than a 45-W bulb?

t'

(k)sin

(k)sin t'

k t'sin t'sin VVk

1-

1-

maxmax

=⇒

=⇒

=⇒∆=∆

ω

ω

ωω

t't'

(k)sin- t" t'- t"

t"sin t'sin ) t'-(sin

:figure by the Guided

t"be timeLet the

1-

⋅=⇒=∴

==

πωπ

ωωωπ

3. In the simple AC circuit shown in the figure, Δv = ΔVmax sin ω t.

(a) If ΔvR = kΔVmax for the first time at t=t’, what is the angular frequency of the source?

(b) What is the next value of t for which ΔvR = kΔVmax?

∆v

t

∆Vmax

k∆Vmax

t'

t=?

4. An AC source has an output rms voltage of Vrms at a frequency of f. The source is connected

across an inductor L. (a) Find the inductive reactance of the circuit. (b) Find the rms current in

the circuit. (c) Find the maximum current in the circuit.

fL 2

V

fL 2

V2I 2I c)(

fL 2

V

|X|

V I b)(

fL 2 L X X a)(

rmsrmsrmsmax

rmsrms

rms

L

ππ

π

πω

=⋅==

==

===

5. A source delivers an AC voltage of the form Δv = Vmax sin (kπ t), where Δv is in volts and t is

in seconds, to a capacitor. The maximum current in the circuit is Imax. (a) Find the rms voltage

of the source. (b) Find the frequency of the source. (c) Find the value of the capacitance.

max

max

max

max

max

max

max

max

maxrms

V

I

k

1 C

V

I

1 C

I

V

C

1

I

V |X|

|I|

|V||X|

I

V X c)(

2

k f k f 2 k b)(

2

V V a)(

π

ω

ω

πππω

=⇒

=⇒

=⇒

=⇒=⇒=

=⇒=⇒=

=

6. An AC source with ΔVmax and frequency f Hz is connected between points a and d in the

figure.

22222max

22222max

22

max

2

CL

2

R

Rmax

CLR

R

CLR

R

]1LCf)2[(RCf)2(

fCR2V

)1LC(RC

CRV

)C

1L(R

RV

)XX(X

|X|V

|XXX|

|X||V|

|XXX

XV| b and a pointsbetween voltagesMaximum a)(

−+⋅=

−+⋅=

−+

⋅=

−+=

++⋅=

++⋅=

ππ

π

ωω

ω

ωω

22222max

22222max

22

max

2

CL

2

R

Rmax

CLR

R

CLR

R

]1LCf)2[(RCf)2(

fCR2V

)1LC(RC

CRV

)C

1L(R

RV

)XX(X

|X|V

|XXX|

|X||V|

|XXX

XV| b and a pointsbetween voltagesMaximum a)(

−+⋅=

−+⋅=

−+

⋅=

−+=

++⋅=

++⋅=

ππ

π

ωω

ω

ωω

(a) Calculate the maximum voltages between points a and b.

(b) Calculate the maximum voltages between points b and c.

(c) Calculate the maximum voltages between points c and d.

R L C

1/(ωC)

ωL

R

22222

2

max

22222

2

max

22

max

CLR

L

22222max

22222max

22

max

2

CL

2

R

Rmax

CLR

R

CLR

R

]1LCf)2[(RCf)2(

LCf)2(V

)1LC(RC

LCV

)C

1L(R

LV

|XXX

XV| c and b pointsbetween voltagesMaximum b)(

]1LCf)2[(RCf)2(

fCR2V

)1LC(RC

CRV

)C

1L(R

RV

)XX(X

|X|V

|XXX|

|X||V|

|XXX

XV| b and a pointsbetween voltagesMaximum a)(

−+⋅=

−+⋅=

−+

⋅=

++⋅=

−+⋅=

−+⋅=

−+

⋅=

−+=

++⋅=

++⋅=

ππ

π

ωω

ω

ωω

ω

ππ

π

ωω

ω

ωω

22222max

22222max

22

max

CLR

C

]1LCf)2[(RCf)2(

1V

)1LC(RC

1V

)C

1L(R

C

1

V

|XXX

XV| d and c pointsbetween voltagesMaximum c)(

−+⋅=

−+⋅=

−+

⋅=

++⋅=

ππ

ωω

ωω

ω

7. An inductor L, a capacitor C, and a resistor R are connected in series. A f-Hz AC source

produces a peak current of Imax in the circuit. (a) Calculate the required peak voltage ∆Vmax.

(b) Determine the phase angle by which the current leads or lags the applied voltage.

.11LC if voltatgeleadsCurrent .11LC if voltagelagsCurret

)CR

1LC( tan

)R

C

1L

( tan figure,upper theFrom

. is V and I betweem angle phase figure,lower thefromseen becan As

V/XI (b)

fC 2

]1LCf)2[(RCf)2(IV

C

)1LC(RCIV

)C

1L(RIV

|XXX|IV

|X|IV

|X||I| | V| IX V (a)

22

21-

1-

22222

maxmax

22222

maxmax

22

maxmax

CLRmaxmax

maxmax

<−>−

−=

−=

=

−+=⇒

−+=⇒

−+=⇒

++=⇒

=⇒

=⇒=

ωω

ω

ω

ωω

φ

φ

π

ππ

ω

ωω

ωω

1/(ωC)

ωL

R

X

φ

X

V

I=V/X

φ φ

)kCR

1LCk( tan

)R

kC

1kL

( tan figure,upper theFrom

. is V and I betweem angle phase figure,lower thefromseen becan As (d)

k c)(

)1LCk(RCk

kCVI

kC

)1LCk(RCk

VI

|X|

|V||I|

X

VI (b)

kC

)1LCk(RCk

)kC

1kL(R |XXX| (a)

21-

1-

22222

max

max

22222

max

max

22222

22

CLR

−=

−=

=

−+=⇒

−+=⇒

=⇒=

−+=

−+=++

φ

φ

ω

9. A sinusoidal voltage ∆v = Vmaxsinkt, where ∆v is in volts and t is in seconds, is applied

to a series RLC circuit. (a) What is the impedance of the circuit? (b) What is the maximum

current? (c) Determine the numerical value for ω in the equation i = Imax sin (ωt − ). (d)

Determine the numerical value for in the equation i = Imax sin (ωt − ).

1/(ωC)

ωL

R

X

V

I=V/X

φ φ

C

L

R

V

R

LLC

1

V

R

LV inductor across voltagesMaximum

0C

1L resonance,At

)C

1L(R

LV

|XXX

XV| inductor across voltagesMaximum d)(

C

L

R

1

R

LLC

1

R

L Q c)(

R

VI

0R

VI 0

C

1L resonance,At

)C

1L(R

VI

C

)1LC(RC

VI

|X|

|V||I|

X

VI (b)

LC

1 frequency Resonant (a)

max

max

0

max

22

max

CLR

L

0

max

max

22

max

max

22

max

max

22222

max

max

0

=

⋅=

⋅=∴

=−

−+

⋅=

++⋅=

===

=⇒

+=⇒=−

−+

=⇒

−+=⇒

=⇒=

=

ω

ωω

ωω

ω

ω

ωω

ωω

ω

ωω

ω

10. A series RLC circuit has components with values L, C, and R,0 Ω, and Δv = ΔVmax sin ωt.

(a) Find the resonant frequency of the circuit.(b) Find the amplitude of the current at the

resonant frequency.(c) Find the Q of the circuit. (d) Find the amplitude of the voltage across

the inductor at resonance.

rmsrms

rms

1

22

1

1

22

1

2

1

2

IV deliveredPower b)(

VN

NV

VN

NV

N

N

V

V a)(

=

⋅=⇒

⋅=⇒=

11. A step-down transformer is used for recharging the batteries of portable electronic devices.

The turns ratio N2/N1 for a particular transformer used in a DVD player is N2 : N1. When used with

Vrms household service, the transformer draws an rms current of Irms from the house outlet. (a) Find

the rms output voltage of the transformer (b) Find the power delivered to the DVD player.

12. The RC high-pass filter shown in the figure below has a resistance R and a capacitance C.

What is the ratio of the amplitude of the output voltage to that of the input voltage for this

filter for a source frequency of f?

2in

out

2inout

2inout

22

inout

2

C

2

R

inout

CR

Rinout

CR

RinRout

fCR)2(1

fCR2

|v |

|v | R

fCR)2(1

fCR2|v | |v |

1CR)(

CR|v | |v|

)C

1(R

R|v | |v|

XX

R|v | |v|

|XX|

|X||v | |v|

XX

Xv v v

π

π

π

π

ω

ω

ω

+=

∆

∆=∴

+⋅∆=∆⇒

+⋅∆=∆⇒

+

⋅∆=∆⇒

+⋅∆=∆⇒

+⋅∆=∆⇒

+⋅∆=∆=∆

1/(ωC)

R

13. A current I is charging a capacitor that has square plates of length L on each side. The

plate separation is d. (a) Find the time rate of change of electric flux between the plates. (b)

Find the displacement current between the plates.

I I

(a))part (from L

IL I

td

E dL I

td

E dA I

td

EA d I

td

d I

td

d I b)(

L

I

dt

dE

dt

dEL I

dt

dEd

d

L I

d

L

d

A C

dt

dECd I

dt

dVC

dt

dQ

Idt

dQ Also,

dt

dEd

dt

dV EdVBut

dt

dVC

dt

dQ CV Q (a)

d

2

0

0

2

d

0

2

d

0d

0d

E0d

E0d

2

0

2

0

2

0

2

00

=⇒

=⇒

=⇒

=⇒

=⇒

Φ=⇒

Φ=

=⇒

=⇒⋅⋅=∴

==

=⇒=∴

=

=⇒=

=⇒=

εε

ε

ε

ε

εε

ε

εε

εε

14. A current I is charging a capacitor that has circular plates R in radius. The plate

separation is d. (a) What is the time rate of increase of electric field between the plates? (b)

What is the magnetic field between the plates r from the center?

I)(I r 2

I B

I r 2 B

I sdB Law sAmpere'

I I b)(

R

I

dt

dE

dt

dER I

dt

dEd

d

R I

d

R

d

A C

dt

dECd I

dt

dVC

dt

dQ

Idt

dQ Also,

dt

dEd

dt

dV EdVBut

dt

dVC

dt

dQ CV Q (a)

d

0

d0

im0

d

2

0

2

0

2

0

2

00

==⇒

=⋅⇒

=⋅⇒

=

=⇒

=⇒⋅⋅=∴

==

=⇒=∴

=

=⇒=

=⇒=

∫

π

µ

µπ

µ

πε

πεπ

ε

πεε

rr

I

I

Id r

B=?

15. In SI units, the electric field in an electromagnetic wave is described by

Ey = Emax sin(kx − ωt). (ω is not given)

(a) Find the amplitude of the corresponding magnetic field oscillations. (b) Find the wavelength λ.

(c) Find the frequency f.

π

πω

πλ

λ

π

2

ck f

ck f 2 ck d)(

k

2

2 k b)(

c

E B

cB E cB E (a)

max

max

maxmax

=⇒

=⇒=

=⇒=

=⇒

=⇒=

16. The speed of an electromagnetic wave traveling in a transparent nonmagnetic

substance is 001/v εκµ= where κ is the dielectric constant of the substance. Determine the

speed of light in water, which has a dielectric constant of 1.78 at optical frequencies.

Solution: Just plug in the numbers!

17. At one location on the Earth, the rms value of the magnetic field caused by solar radiation

is Brms. (a) Calculate the rms electric field due to solar radiation. (b) Calculate the average

energy density of the solar component of electromagnetic radiation at this location. (c)

Calculate the average magnitude of the Poynting vector for the Sun's radiation. (d) Assuming

that the average magnitude of the Poynting vector for solar radiation at the surface of the

Earth is Searth, compare your result in part (c) with this value.

100%S

S isanwser The (d)

)uc( B

c S

Bc

S

cB)(E cB

S

EB S BE

1 S c)(

B 1

B 2

1 B

1

2

1

)1

(c B 2

1 B

1

2

1

B 2

1 Bc

2

1

(a))part (from B 2

1 )(cB

2

1

B 2

1 E

2

1

B2

1 E

2

1 u

B2

1 E

2

1 uu u (b)

cB E cB E (a)

earth

avg

0

2

rms

avg

0

2

avg

0

2

00

2

rms

0

2

rms

0

2

rms

0

00

2

rms

0

2

rms

00

0

2

rms

0

2

rms

2

0

2

rms

0

2

rms0

2

rms

0

2

rms0

2

0

2

0

2

0

2

0BE

rmsrms

×

><==⇒

><=⇒

==⇒

=⇒×=

=

+=

=+⋅=

+=

+=

+=

><+><>=∴<

+=+=

=⇒=

µ

µ

µ

µµ

µ

µµ

µεµµεε

µε

µε

µε

µε

µε

rrr

18. A radio wave transmits W of power per unit area. A flat surface of area A is

perpendicular to the direction of propagation of the wave. Assuming the surface is a perfect

absorber, calculate the radiation pressure on it.

avg

avg

S as same theis with intensity, just the isgiven W The :Note

c

W

c

S P absornbed, completely isradiation theSince

=

=

19. A power W laser beam of diameter D is reflected at normal incidence by a perfectly

reflecting mirror. Calculate the radiation pressure on the mirror.

2

2

22

avg

avg

cD

8W

c

D

4W2

P

D

4W

)2

D(

W

A

W I S

c

2S P reflected, isradiation theSince

ππ

ππ

=

⋅

=∴

====

=

20. What is the wavelength of electromagnetic wave in free space that have frequency f?

f

c fc =⇒= λλ

20.