Electricity%20and%20Magnetism

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Chapter 26
Electricity and Magnetism Electric Field
Continuous Charge Distribution
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Find electric field at point P.
Continuous Charge Distribution
P
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Electric Field Continuous Charge Distribution
Electric field
In this situation, the system of charges can be
modeled as continuous
The system of closely spaced charges is
equivalent to a total charge that is continuously
distributed along some line, over some surface,
or throughout some volume
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Electric Field Continuous Charge Distribution
The total electric charge is Q. What is the
electric field at point P ?
P
P
linear
volume
Q
P
Q
surface
Q
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Continuous Charge Distribution Charge Density
The total electric charge is Q.


Volume V
Linear, length L
Q
Amount of charge in a small volume dl
Q
Amount of charge in a small volume dV
Linear charge density
Volume charge density

Surface, area A
Amount of charge in a small volume dA
Q
Surface charge density
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Electric Field Continuous Charge Distribution

• Procedure
• Divide the charge distribution into small
  elements, each of which contains ?q
• Calculate the electric field due to one of these
  elements at point P
• Evaluate the total field by summing the
  contributions of all the charge elements
• Symmetry take advantage of any symmetry to
  simplify calculations

For the individual charge elements
Because the charge distribution is continuous
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Electric Field Symmetry
no symmetry
no symmetry
no symmetry
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Electric Field Symmetry
Electric field
The symmetry gives us the direction of resultant
electric field
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Electric Field Continuous Charge Distribution
What is the electric field at point P ?
- linear charge density
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- linear charge density
What is the electric field at point P ?
1. Symmetry determines the direction of the
electric field.
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- linear charge density
What is the electric field at point P ?
2. Divide the charge distribution into small
elements, each of which contains ?q
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- linear charge density
What is the electric field at point P ?
3. Evaluate the total field by summing the
contributions of all the charge elements ?q
Replace Sum by Integral
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- linear charge density
What is the electric field at point P ?
4. Evaluate the integral
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- linear charge density
What is the electric field at point P ?
From symmetry
Replace the Sum by Integral
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- surface charge density
What is the electric field at point P ?
1. Symmetry determines the direction of the
electric field.
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- surface charge density
What is the electric field at point P ?
2. Divide the charge distribution into small
elements, each of which contains ?q 3. Evaluate
the total field by summing the contributions of
all the charge elements ?q
Approach A straightforward
at point (x,y)
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Approach A
- surface charge density
What is the electric field at point P ?
Replace the Sum by Integral
at point
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Approach A
- surface charge density
What is the electric field at point P ?
Evaluate the Integral
at point
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Approach B
- surface charge density
What is the electric field at point P ?
Charge of the ring
Linear density of the ring
Ring of width
Length of the ring
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Electric Field Symmetry
no symmetry
no symmetry
no symmetry
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no symmetry, but we know and
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• no symmetry
• We do not know the direction of electric field
  at point P
• We need to find x and y component of electric
  field

Then
Then
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• no symmetry
• We do not know the direction of electric field
  at point P
• We need to find x and y component of electric
  field

the symmetry tells us that one of the component
is 0, so we do not need to calculate it.
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Important Example
Find electric field
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Important Example
Find electric field
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Important Example
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Motion of Charged Particle
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Motion of Charged Particle

• When a charged particle is placed in an electric
  field, it experiences an electrical force
• If this is the only force on the particle, it
  must be the net force
• The net force will cause the particle to
  accelerate according to Newtons second law

- Coulombs law
- Newtons second law
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Motion of Charged Particle
What is the final velocity?
- Coulombs law
- Newtons second law
Motion in x with constant velocity
Motion in x with constant acceleration
- travel time
After time t the velocity in y direction becomes
then