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W02D1Electric Dipoles and Continuous

Charge Distributions

Announcements

Math Review Tuesday Tues Feb 14 from 9-11 pm in 32-082

PS 1 due Tuesday Tues Feb 14 at 9 pm in boxes outside 32-082 or 26-152

W02D2 Reading Assignment Course Notes: Chapter Course Notes: Sections 4.1-4.2, 4.7

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Outline

Electric DipolesForce and Torque on DipoleContinuous Charge Distributions

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Nature Likes to Make Dipoles

http://youtu.be/EMj10YIjkaY

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Demonstration: Dipole in a Van de Graaff

Generator D22

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Concept Question: Dipole in Non-Uniform Field

A dipole sits in a non-uniform electric field E

E

Due to the electric field this dipole will feel:

1. force but no torque

2. no force but a torque

3. both a force and a torque

4. neither a force nor a torque

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Concept Question Answer: Non-Uniform Field

Because the field is non-uniform, the forces on the two equal but opposite point charges do not cancel.

As always, the dipole wants to rotate to align with the field – there is a torque on the dipole as well

Answer: 3. both force and torqueE

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Continuous Charge Distributions

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?P E

V

Continuous Charge Distributions

Break distribution into parts:

E field at P due to q

Superposition:

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Continuous Sources: Charge Density

Length L

L

w

L

R

L

dLdQ

dAdQ

dVdQ

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Group Problem: Charge Densities

A solid cylinder, of length L and radius R, is uniformly charged with total charge Q.

(a)What is the volume charge density ρ?

(b)What is the linear charge density λ?

(c)What is the relationship between these two densities ρ and λ?

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Examples of Continuous Sources: Finite Line of Charge

L

Q

Length L

L dLdQ

E field on perpendicular

bisector

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Examples of Continuous Sources: Finite Line of Charge

L

Q

Length L

L

dLdQ

E field off axis

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Examples of Continuous Sources: Finite Line of Charge

L

Q

Length L

L

dLdQ

Grass seeds of total E field

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Concept Question Electric Field of a Rod

A rod of length L lies along the x-axis with its left end at the origin. The rod has a uniform charge density λ. Which of the following expressions best describes the electric field at the point P

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Concept Question Electric Field of a Rod: Answer

A rod of length L lies along the x-axis with its left end at the origin. The rod has a uniform charge density λ. Which of the following expressions best describes the electric field at the point P

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Group Problem: Line of Charge

Point P lies on perpendicular bisector of uniformly charged line of length L, a distance s away. The charge on the line is Q. Find an integral expression for the direction and magnitude of the electric field at P.

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r

2

L

2

L

xd

x

xddq s

22 xsr

P

j

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Hint on Line of Charge Group Problem

Typically give the integration variable (x’) a “primed” variable name. ALSO: Difficult integral (trig. sub.)

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E Field from Line of Charge

Limits: s >> L (far away) and s << L (close)

Looks like the E field of a point charge if we are far away

Looks like E field of an infinite charged line if we are close

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Examples of Continuous Sources: Ring of Charge

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Q

R

dLdQ

E field on the axis of the ring

of charge

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Examples of Continuous Sources: Ring of Charge

2

Q

R

dLdQ

E field off axis and grass seeds plot

A uniformly charged ring of radius a has total charge Q. Which of the following expressions best describes the electric field at the point P located at the center of the ring?

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Concept Question Electric Field of a Ring

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Concept Question Electric Field of a Ring: Answer

A uniformly charged ring of radius a has total charge Q. Which of the following expressions best describes the electric field at the point P located at the center of the ring?

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Demonstration Problem: Ring of Charge

A ring of radius a is uniformly charged with total charge Q. Find the direction and magnitude of the electric field at the point P lying a distance x from the center of the ring along the axis of symmetry of the ring.

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Ring of Charge

Symmetry!1) Think about it

2) Define Variables

22 xar

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Ring of Charge

3) Write Equation22 xar

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Ring of Charge

4) Integrate

This particular problem is a very special case because

everything except dq is constant, and

22 xar

Q

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Ring of Charge

5) Clean Up

6) Check Limit

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Group Problem: Uniformly Charged Disk

P on axis of disk of charge, x from center

Radius R, charge density .

Find E at P

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Disk: Two Important Limits

Limits: x >> R (far) and x << R (close)

Looks like E of a point charge far away

Looks like E field of an infinite charged plane close up

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Scaling: E for Plane is Constant

1) Dipole: E falls off like 1/r3

2) Point charge: E falls off like 1/r2

3) Line of charge: E falls off like 1/r

4) Plane of charge: E constant