陳銘堯老師普通物理下
HW-06
1. (a) Find the magnetic field at the center of a regular n-sided polygon, carrying a
steady current I. Let R be the distance from the center to any side. (b) Check that
your answer to (a) reduces to the field at the center of a circular loop in the limit n ->
∞.
2. A short solenoid carries a current I and has n turns/meter. Show, by adding up
(integration) the magnetic field from each coil, that B = 0In(cos1 + cos2)/2 at any
point on the axis of the solenoid. Note that 1 > /2 and 2 < /2 for the point on the
left outside the solenoid, but 1 < /2 and 2 > /2 for the point on the right outside.
Check how quickly the field drops off outside the solenoid.
1
1
2
2
R
R
L
3. (a) Find the magnetic field inside an infinite current carrying sheet of thickness
2D with a uniform current density j. (b) Find the magnetic field of an infinitely long
coaxial cable with currents I in opposite directions in the inner and outer conductors.
(You have to find the magnetic field inside the conductors.) Assume the currents are
distributed uniformly throughout the cross sections of the conductors.
c
I
I
b
a
4. A toroidal coil of N turns is wound on a form of nonmagnetic material having a
rectangular cross section with height H. The inner radius of the toroid is a, and the
outer radius is b. We can decompose the magnetic field into Br (radial), Bz (vertical)
and B (azimuthal) components. It can be shown form the Biot-Savart law that both
Br and Bz vanish, as long as (1) any current element I d s⃑ of the winding has no 
component, and (2) for each I d s⃑ at  there is a corresponding current element I d s⃑′ at
-  by rotational symmetry to cancel Br and Bz. (Why?)
(a) Find B inside and outside the toroid. (b) Show that the result of (a) is similar to
the expression for a long solenoid when a >> H.
z
b
a
 Bz
H y
B
Br
x
Top View
y
- 
I d s⃑
I d s⃑′
x
5. The term I d s⃑ in the Biot-Savart law can be rewritten as ⃑j dv, since I = j da. So,
the law becomes a volume integral:
⃑⃑(r⃑) =
B
μo
∫
4π
⃑j(r⃑⃑′ )×(r⃑⃑−r⃑⃑′)
|r⃑⃑−r⃑⃑′|3
dv′.
⃑⃑ ∙ B
⃑⃑ = 0. [Hint: Apply divergence (∇
⃑⃑) to the integrand and use the
Show that ∇
⃑⃑⃑⃑⃑ × ⃑f ) − ⃑f ∙ (∇
⃑⃑⃑⃑⃑ × ⃑g⃑))]
formula: ⃑∇⃑ ∙ (f⃑ × ⃑g⃑) = ⃑g⃑ ∙ (∇