Page 353 - Intro to Tensor Calculus

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EXERCISE 2.6

I 1. Find the field lines and equipotential curves associated with a positive charge q located at (−a, 0) and
a positive charge q located at (a, 0). The field lines are illustrated in the figure 2.6-7.

Figure 2.6-7. Lines of electric force between two charges of the same sign.

I 2. Calculate the lines of force and equipotential curves associated with the electric field
~
~
E = E(x, y)= 2y b e 1 +2x b e 2 . Sketch the lines of force and equipotential curves. Put arrows on the lines of
force to show direction of the field lines.
I 3. A right circular cone is defined by

x = u sin θ 0 cos φ, y = u sin θ 0 sin φ, z = u cos θ 0

A
with 0 ≤ φ ≤ 2π and u ≥ 0. Show the solid angle subtended by this cone is Ω = r 2 =2π(1 − cos θ 0 ).
I 4. A charge +q is located at the point (0,a) and a charge −q is located at the point (0, −a). Show that
1 −2aq
~
~
the electric force E at the position (x, 0), where x> a is E = b e 2 .
2 3/2
2
4π 0 (a + x )
2
2
2
I 5. Let the circle x + y = a carry a line charge λ . Show the electric field at the point (0, 0,z)is
∗
1 λ az(2π) b e 3
∗
~
E = .
2
2 3/2
4π 0 (a + z )
I 6. Use superposition to find the electric field associated with two infinite parallel plane sheets each
∗
carrying an equal but opposite sign surface charge density µ . Find the field between the planes and outside
µ ∗
of each plane. Hint: Fields are of magnitude ± and perpendicular to plates.
2 0
ZZZ ~
~
~
~
I 7. For a volume current J the Biot-Savart law gives B = µ 0 J × b e r dτ. Show that ∇· B =0.
4π V r 2
~ r r ~
~
Hint: Let b e r = and consider ∇· (J × ). Then use numbers 13 and 10 of the appendix C. Also note that
r r 3
~
~
∇× J = 0 because J does not depend upon position.