W15d1 poynting vector and energy flow
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W15D1: Poynting Vector and Energy Flow. Today ’ s Readings: Course Notes: Sections 13.6, 13.12.3-13.12.4 . Final Math Review Week 15 Tues from 9-11 pm in 32-082 Final Exam Monday Morning May 20 from 9 am-12 noon Johnson Athletic Center Track 2 nd floor . Announcements. Outline.

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W15D1: Poynting Vector and Energy Flow

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W15d1 poynting vector and energy flow

W15D1:Poynting Vector and Energy Flow

Today’s Readings:

Course Notes: Sections 13.6, 13.12.3-13.12.4


Announcements

Final Math Review

Week 15 Tues from 9-11 pm in 32-082

Final Exam Monday Morning

May 20 from 9 am-12 noon

Johnson Athletic Center Track 2nd floor

Announcements


Outline

Outline

Poynting Vector and Energy Flow

Examples


Maxwell s equations

Maxwell’s Equations


Electromagnetism review

Electromagnetism Review

  • Conservation of charge:

  • E and B fields exert forces on (moving) electric charges:

  • Energy stored in electric and magnetic fields


Energy flow

Energy Flow


Poynting vector

Poynting Vector

Power per unit area:

Poynting vector

Power through a surface


Group problem resistor power

Group Problem: Resistor Power

a

I

Consider the above cylindrical resistor with resistance R and current Iflowing as shown.

  • What are the electric and magnetic fields on the surface of the resistor?

  • Calculate the flux of the Poynting vector through the surface of the resistor (power) in terms of the electric and magnetic fields.

  • Express your answer to part b) in terms of the current I and resistance R. Does your answer make sense?

L


Energy flow resistor

Energy Flow: Resistor

On surface of resistor direction is INWARD


Energy flow capacitor

Energy Flow: Capacitor

What is the magnetic field?


Displacement current

Displacement Current

So we had to modify Ampere’s Law:


Sign conventions right hand rule

Sign Conventions: Right Hand Rule

Integration direction counter clockwise for line integral requires that unit normal points out page for surface integral.

Current positive out of page. Negative into page.

Electric flux positive out of page, negative into page.

B field clockwise, line integral negative.

B field counterclockwise, line integral positive.


Concept questions poynting vector

Concept Questions:Poynting Vector


Concept question capacitor

Concept Question: Capacitor

The figures above show a side and top view of a capacitor with charge Q and electric and magnetic fields E and B at time t. At this time the charge Q is:

  • Increasing in time

  • Constant in time.

  • Decreasing in time.

  • Not enough information given to determine how Q is changing.


Concept q answer capacitor

Concept Q. Answer: Capacitor

Answer: 3. The charge Q is decreasing in time

Use the Ampere-Maxwell Law. Choose positive unit normal out of plane. Because the magnetic field points clockwise line integral is negative hence positive electric flux (out of the plane of the figure on the right) must be decreasing. Hence E is decreasing. Thus Q must be decreasing, since E is proportional to Q.


Concept question capacitor1

Concept Question: Capacitor

The figures above show a side and top view of a capacitor with charge Q and electric and magnetic fields E and B at time t. At this time the energy stored in the electric field is:

  • Increasing in

  • Constant in time.

  • Decreasing in time.


Concept q answer capacitor1

Concept Q. Answer: Capacitor

Answer: 1. The the energy stored in the electric field is increasing in time

The direction of the Poynting Flux S (= E x B) inside the capacitor is inward. Therefore electromagnetic energy is flowing inward, and the energy in the electric field inside is increasing.


Group problem capacitor

Group Problem: Capacitor

A circular capacitor of spacing d and radius R is in a circuit carrying the steady current i shown. Ignore edge effects. At time t = 0 it is uncharged

  • Find the electric field E(t) at A (mag. & dir.)

  • Find the magnetic field B(t) at A (mag. & dir.)

  • Find the Poynting vector S(t) at A (mag. & dir.)

  • What is the flux of the Poynting vector into/out of the capacitor?

  • How does this compare to the time derivative of the energy stored in the electric field?

  • Does this make sense?


Another look at inductance

Another look at Inductance


Faraday inductors

L

I

Faraday & Inductors


Concept question inductor

Concept Question: Inductor

The figures above show a side and top view of a solenoid carrying current I with electric and magnetic fields E and B at time t. The current I is

  • increasing in time.

  • constant in time.

  • decreasing in time.


Concept question answer inductor

Concept Question Answer: Inductor

Answer: 3. The current I is decreasing in time

Use Faraday’s law. Choose positive unit normal out of plane. Because the electric field points counterclockwise line integral is positive, therefore the positive magnetic flux must be decreasing (out of the plane of the figure on the right). Hence B is decreasing. Thus I must be decreasing, since B is proportional to I.


Concept question inductor1

Concept Question: Inductor

The figures above show a side and top view of a solenoid carrying current I with electric and magnetic fields E and B at time t. The energy stored in the magnetic field is

  • Increasing in time

  • Constant in time.

  • Decreasing in time.


Concept question answer inductor1

Concept Question Answer: Inductor

Answer: 3. The energy stored in the magnetic field is decreasing in time.

The Poynting Flux S (= E x B) inside the solenoid is directed outward from the center of the solenoid. Therefore EM energy is flowing outward, and the energy stored in the magnetic field inside is decreasing.


Group problem inductor

Group Problem: Inductor

A solenoid of radius a and length h has an increasing current I(t) as pictured.

Consider a point D inside the solenoid at radius r (r < a). Ignore edge effects.

  • Find the magnetic field B(t) at D (dir. and mag.)

  • Find the electric field E(t) at D (dir. and mag.)

  • Find the Poynting vector S(t) at D (dir. and mag.)

  • What is the flux of the Poynting vector into/out of the inductor?

  • How does this compare to the time derivative of the energy stored in the magnetic field?

  • Does this make sense to you?


Energy flow inductor

Energy Flow: Inductor

Direction on surface of inductor with increasing current is INWARD


Energy flow inductor1

Energy Flow: Inductor

Direction on surface of inductor with decreasing current is OUTWARD


Power energy in circuit elements

Power & Energy in Circuit Elements

Dissipates

Power

Store

Energy

POWER

When

(dis)charging


Poynting vector and em waves

Poynting Vector and EM Waves


Energy in em waves

Energy in EM Waves

Energy densities:

Consider cylinder:

What is the energy flow per unit area?


Poynting vector and intensity

Poynting Vector and Intensity

Direction of energy flow = direction of wave propagation

units: Joules per square meter per sec

Intensity I:


Momentum radiation pressure

Momentum & Radiation Pressure

EM waves transport energy:

They also transport momentum:

They exert a pressure:

This is only for hitting an absorbing surface. For hitting a perfectly reflecting surface the values are doubled:


W15d1 poynting vector and energy flow

Group Problem: Radiation

A light bulb puts out 100 W of electromagnetic radiation. What is the time-average intensity of radiation from this light bulb at a distance of one meter from the bulb? What are the maximum values of electric and magnetic fields, E and B, at this same distance from the bulb? What is the pressure this radiation will exert on a very small perfectly conducting plate at 1 meter. For simplicity, you may assume the radiation is a plane wave of wavelength λ.


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