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LECTURE 4 slide 1

Lecture 4

Vector Superposition

The Field of Standard ChargeDistributions

Sections: 2.4, 2.5

Homework: D2.2, D2.5, D2.6; 2.17, 2.18, 2.20, 2.25

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LECTURE 4 slide 2

E-field of Multiple Sources

superposition holds in a linear medium

superposition means adding the individual contributions of sources

vector superposition implies adding/subtracting vectors

the E-field is a vector

The ESF of multiple charges at any point is a vectorial

sum of the fields created by each individual charge.

i

i

= E E

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LECTURE 4 slide 3

E-field of a Point Charge Reminder

a point charge at the origin (spherical coordinates)

2

1

4r

Q

r=E a

a point charge at position r

2

1 ( )( )

4 | |R

Q

=

rE r a

r r or

3

( ) ( )( )

4 | |

Q

=

r r rE r

r r

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LECTURE 4 slide 4

E-field of Multiple Point Charges 1

21

1 ( )( )

4 | |

Nn n

n

nn

Q

=

=

r

E r ar r

Examples with 2 charges:

1 2Q Q= 2Q Q= E

1r

2 1r r=

1E2E

E

2E

1E

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LECTURE 4 slide 5

E-field of Multiple Point Charges 2

1 1 1 11 2 2 2 3/ 2

1 1 1

( ) ( ) ( )( , , )

4 | ( ) ( ) ( ) |

y zQ x x y y z z x y z

x x y y z z + + = + +

a a aE

2 2 2 2

2 2 2 2 3/ 22 2 2

( ) ( ) ( )

( , , ) 4 | ( ) ( ) ( ) |

x y zQ x x y y z z

x y z x x y y z z

+ +

= + +

a a a

E

1 2( , , ) ( , , ) ( , , )x y z x y z x y z= +E E E

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LECTURE 4 slide 6

E-field of a Uniform Line Charge 1

The line charge is

represented by an

infinite number of

infinitesimal line

elements of charge

ldQ dz =

The field of the linecharge is a super-

position of the fields

created by each

linear charge

element dQ.

dz ldQ dz =

( , , 0 )

z

x y0

L

1R

2

1

2

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LECTURE 4 slide 7

E-field of a Uniform Line Charge 2

The contribution of each line element is

3

( )

4 | |

ldzd

=

r rE

r r

, =0

, 0, 0 2

zz

z

=

= =