25. Consider an infinitesimal section of the rod of length dx, a distance x from the left end, as shown in the

diagram below. It contains charge dq = λ dx and is a distance r from P . The magnitude of the field it
produces at P is given by

dE =

1

4πε

0

λ dx

r

2

.

The x component is

dE

x

=

−

1

4πε

0

λ dx

r

2

sin θ

and the y component is

dE

y

=

−

1

4πε

0

λ dx

r

2

cos θ .

x

y

dq

x

..

..

..

..

.

..

..

..

..

..

..

..

..

..

..

..

..

.

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

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..

.

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

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..

..

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.

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.

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.

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.

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..

..

..

R

r

θ

P

•............

..................

..................

.................

..................

.................

...........................

......

.....................

.d 

E

We use θ as the variable of integration and substitute r = R/ cos θ, x = R tan θ and dx = (R/ cos

2

θ) dθ.

The limits of integration are 0 and π/2 rad. Thus,

E

x

=

−

λ

4πε

0

R

π/2

0

sin θ dθ =

λ

4πε

0

R

cos θ



π/2

0

=

−

λ

4πε

0

R

and

E

y

=

−

λ

4πε

0

R

π/2

0

cos θ dθ =

−

λ

4πε

0

R

sin θ



π/2

0

=

−

λ

4πε

0

R

.

We notice that E

x

= E

y

no matter what the value of R. Thus, 

E makes an angle of 45

◦

with the rod

for all values of R.

Document Outline

• Main Menu
• Chapter 1 Measurement
• Chapter 2 Motion Along a Straight Line
• Chapter 3 Vectors
• Chapter 4 Motion in Two and Three Dimensions
• Chapter 5 Force and Motion I
• Chapter 6 Force and Motion II
• Chapter 7 Kinetic Energy and Work
• Chapter 8 Potential Energy and Conservation of Energy
• Chapter 9 Systems of Particles
• Chapter 10 Collisions
• Chapter 11 Rotation
• Chapter 12 Rolling, Torque, and Angular Momentum
• Chapter 13 Equilibrium and Elasticity
• Chapter 14 Gravitation
• Chapter 15 Fluids
• Chapter 16 Oscillations
• Chapter 17 Waves—I
• Chapter 18 Waves—II
• Chapter 19 Temperature, Heat, and the First Law of Thermodynamics
• Chapter 20 The Kinetic Theory of Gases
• Chapter 21 Entropy and the Second Law of Thermodynamics
• Chapter 22 Electric Charge
• Chapter 23 Electric Fields
  • 23.1 - 23.10
    • 23.1
    • 23.2
    • 23.3
    • 23.4
    • 23.5
    • 23.6
    • 23.7
    • 23.8
    • 23.9
    • 23.10
  • 23.11 - 23.20
    • 23.11
    • 23.12
    • 23.13
    • 23.14
    • 23.15
    • 23.16
    • 23.17
    • 23.18
    • 23.19
    • 23.20
  • 23.21 - 23.30
    • 23.21
    • 23.22
    • 23.23
    • 23.24
    • 23.25
    • 23.26
    • 23.27
    • 23.28
    • 23.29
    • 23.30
  • 23.31 - 23.40
    • 23.31
    • 23.32
    • 23.33
    • 23.34
    • 23.35
    • 23.36
    • 23.37
    • 23.38
    • 23.39
    • 23.40
  • 23.41 - 23.50
    • 23.41
    • 23.42
    • 23.43
    • 23.44
    • 23.45
    • 23.46
    • 23.47
    • 23.48
    • 23.49
    • 23.50
  • 23.51 - 23.60
    • 23.51
    • 23.52
    • 23.53
    • 23.54
    • 23.55
    • 23.56
    • 23.57
    • 23.58
    • 23.59
    • 23.60
  • 23.61 - 23.68
    • 23.61
    • 23.62
    • 23.63
    • 23.64
    • 23.65
    • 23.66
    • 23.67
    • 23.68
• Chapter 24 Gauss’ Law
• Chapter 25 Electric Potential
• Chapter 26 Capacitance
• Chapter 27 Current and Resistance
• Chapter 28 Circuits
• Chapter 29 Magnetic Fields
• Chapter 30 Magnetic Fields Due to Currents
• Chapter 31 Induction and Inductance
• Chapter 32 Magnetism of Matter: Maxwell’s Equation
• Chapter 33 Electromagnetic Oscillations and Alternating Current
• Chapter 34 Electromagnetic Waves
• Chapter 35 Images
• Chapter 36 Interference
• Chapter 37 Diffraction
• Chapter 38 Special Theory of Relativity
• Chapter 39 Photons and Matter Waves
• Chapter 40 More About Matter Waves
• Chapter 41 All About Atoms
• Chapter 42 Conduction of Electricity in Solids
• Chapter 43 Nuclear Physics
• Chapter 44 Energy from the Nucleus
• Chapter 45 Quarks, Leptons, and the Big Bang