8—Multivariable Calculus 2098.26 In problem2.26you computed the electric potential at large distances from a set of three charges,
− 2 qat the origin and+qatz=±a(r a). The result was
V =
kqa^2
r^3
P 2 (cosθ)
whereP 2 (cosθ)is the second order Legendre polynomial. Compute the electric field from this potential,
E~=−∇V. And sketch it of course.
8.27 Compute the area of an ellipse having semi-major and semi-minor axesaandb. Compare your
result to that of Eq. (8.35). Ans:πab
8.28 Two equal point chargesqare placed atz=±a. The origin is a point of equilibrium;E~ = 0
there. (a) Compute the potential near the origin, writingV in terms of powers ofx,y, andznear
there, carrying the powers high enough to describe the nature of the equilibrium point. IsV maximum,
minimum, or saddle point there? It will be easier if you carry the calculation as far as possible using
vector notation, such as|~r−azˆ|=
√
(~r−azˆ)^2 , andr a.
(b) Write your result forV near the origin in spherical coordinates also.
Ans: 4 π^2 q 0 a
[
1 +r2
a^2( 3
2 cos(^2) θ− 1
2
)]
8.29 When currentI flows through a resistanceRthe heat produced is I^2 R. Two terminals are
connected in parallel by three resistors having resistanceR 1 ,R 2 , andR 3. Given that the total current
is divided asI=I 1 +I 2 +I 3 , show that the condition that the total heat generated is a minimum
leads to the relationI 1 R 1 =I 2 R 2 =I 3 R 3. You can easily do problem8.8by eliminating a coordinate
then doing a derivative. Here it’s starting to get sufficiently complex that you should use Lagrange
multipliers. Doesλhave any significance this time?
8.30 Given a right circular cylinder of volumeV, what radius and height will provide the minimum
total area for the cylinder. Ans:r= (V/ 2 π)^1 /^3 ,h= 2r
8.31 Sometimes the derivative isn’t zero at a maximum or a minimum. Also, there are two types
of maxima and minima; local and global. The former is one that is max or min in the immediate
neighborhood of a point and the latter is biggest or smallest over the entire domain of the function.
Examine these functions for maxima and minima both inside the domains and on the boundary.
|x|, (− 1 ≤x≤+2)
T 0
(
x^2 −y^2
)
/a^2 , (−a≤x≤a, −a≤y≤a)
V 0 (r^2 /R^2 )P 2 (cosθ), (r≤R, 3 dimensions)
8.32 In Eq. (8.39) it is more common to specifyN andβ= 1/kT, the Lagrange multiplier, than it
is to specifyN andE, the total energy. Pick three energies,E`, to be 1, 2, and 3 electron volts.
(a) What is the average energy,E/N, asβ→∞(T→ 0 )?
(b) What is the average energy asβ→ 0?
(c) What aren 1 ,n 2 , andn 3 in these two cases?
8.33 (a) Find the gradient ofV, whereV =V 0 (x^2 +y^2 +z^2 )a−^2 e−
√
x^2 +y^2 +z^2 /a. (b) Find thegradient ofV, whereV =V 0 (x+y+z)a−^1 e−(x+y+z)/a.
8.34 A billiard ball of radiusRis suspended in space and is held rigidly in position. Very small pellets
are thrown at it and the scattering from the surface is completely elastic, with no friction. Compute