Help in explaining what the book said in fields of a moving point charge.

In summary: Boyle's book, "Electric Field: The Energetic Force that Shapes the Universe" provides a solution to the problem. In summary, when a charge is moving, the electric field surrounding it is stretched out in all directions rather than being concentrated in a single direction. This is due to the sin^2\theta in the denominator of the equation for the electric field.
  • #1
yungman
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My questions are what the book said after working out the solution that I have no issue of finding. The original question is to find E and B at a field point pointed by r due to a moving point charge pointed by w(t_r) at retard time moving at a constant velocity v.


For constant velocity:

[tex] \vec E_{(\vec r, t)} = \frac q {4\pi\epsilon_0}\frac {\eta}{(\vec{\eta}\cdot\vec u)^3} [(c^2-v^2)\vec u ] [/tex]

Where

[tex] \vec{\eta} = \vec r -\vec w(t_r), \; \vec u=c\hat{\eta}-\vec v\;,\; \vec v = \frac { d \vec w(t_r)}{dt_r}[/tex]

I have no problem finding the answer:

[tex] \vec E_{(\vec r, t)} = \frac q {4\pi\epsilon_0} \frac {1-\frac{v^2}{c^2}} {\left (1-\frac{v^2}{c^2} sin^2\theta \right )^{\frac 3 2}} \frac {\hat R}{R^2} \;\hbox { where }\; \vec R = \vec r –t\vec v [/tex]





Below are the three statements directly quoted by the book and I have no idea what they mean:

[BOOK]

1) Because of the [tex] sin^2\theta [/tex] in the denominator, the field of a fast-moving charge is flattened out like a pancake in the direction perpendicular to the motion.

2) In the forward and backward directions, E is reduced by a factor [tex] (1 - \frac {v^2}{c^2} ) [/tex] relative to the field of a charge at rest;

3) In the perpendicular direction it is enhanced by a factor [tex]\frac 1 {\sqrt { \left (1 - \frac {v^2}{c^2}\right )}} [/tex]

[END BOOK]








Also in finding B


[tex] \vec B =\frac 1 c \hat{\eta}\times \vec E = \frac 1 c \hat{\eta} \times \frac {q\left ( 1-\frac {v^2}{c^2}\right )}{4\pi\epsilon_0\left ( 1-\frac {v^2}{c^2} sin^2\theta \right )^{\frac 3 2}} \frac {\vec R}{R^3}[/tex]

In two different ways, I get different answer.


1)
[tex] \hat {\eta} \times \vec R = \frac {1}{\eta} [(\vec r -t_r \vec v)\times(c\frac{\vec{\eta}}{\eta}-t\vec v)]=\frac {1}{\eta}[-\frac {ct_r}{\eta} \vec r\times \vec v -t(\vec r \times \vec v) -\frac{ct_r}{\eta}(\vec v\times \vec r)]=-\frac {t} {\eta} (\vec r \times \vec v)[/tex]

[tex] \Rightarrow\;\vec B = \frac 1 c \frac {q\left ( 1-\frac {v^2}{c^2} \right )}{4\pi\epsilon_0\eta \left ( 1-\frac {v^2}{c^2} sin^2\theta\right )^{\frac 3 2}R^3}\;[ t (\vec v \times \vec r)] [/tex]



2)
[tex]\hat{\eta}=\frac{\vec r-t_r\vec v}{\eta}=\frac {(\vec-t\vec v)+(t-t_r)\vec v}{\eta}=\frac {\vec R}{\eta} + \frac {\vec v}{c} \Rightarrow\; \vec B = \frac 1 c \frac {q\left ( 1-\frac {v^2}{c^2} \right )}{4\pi\epsilon_0\eta \left ( 1-\frac {v^2}{c^2} sin^2\theta\right )^{\frac 3 2}R^3} \;(\vec v \times \vec r) [/tex]


Notice a difference of t between the two method? I cannot resolve this.
 
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  • #3
Anyone please, even if you don't have the answer, give me a link or point me to materials that explain more on this. I have been stuck for like two weeks regarding to the electric field of the moving charge when moving at high rate of speed.

Thanks

Alan
 

1. How does a moving point charge affect electric and magnetic fields?

When a point charge moves, it creates electric and magnetic fields in its surrounding space. The electric field is created by the charge's electric force, while the magnetic field is created by the charge's movement.

2. What is the definition of a moving point charge?

A moving point charge is an electric charge that is in motion. This means that it has both magnitude and direction, and it creates electric and magnetic fields as it moves.

3. How can the electric and magnetic fields of a moving point charge be calculated?

The electric and magnetic fields of a moving point charge can be calculated using mathematical equations, such as Coulomb's law and the Biot-Savart law. These equations take into account the charge's magnitude, velocity, and distance from other charges or objects.

4. What are the differences between the electric and magnetic fields of a moving point charge?

The electric field of a moving point charge is affected by the charge's magnitude and the distance from other charges, while the magnetic field is affected by the charge's velocity and the distance from other moving charges. Additionally, the electric field is a conservative field, meaning its work is independent of the path taken, while the magnetic field is a non-conservative field.

5. How do the electric and magnetic fields of a moving point charge interact with each other?

The electric and magnetic fields of a moving point charge can interact with each other through electromagnetic induction. This occurs when a changing magnetic field created by the moving charge induces an electric field, and vice versa. This phenomenon is the basis for many technologies, such as generators and motors.

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