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Homework answers / question archive /  Evaluate the following integrals:a) The integral of (r^2 - 3ar + a^2)delta(r-a)d*tau where a is a fixed vector, a is its magnitude, and the integration is carried out over the whole space; b) The integral (r^4 + r^2 (r*c) + c^4) delta (r-c)d*tau where V is the volume of a cylinder of radius 6 about the z-axis, the bottom at z = -13, the tope at z = 7, the vector c = 5x + 3y + 2z and c is its magnitude

 Evaluate the following integrals:a) The integral of (r^2 - 3ar + a^2)delta(r-a)d*tau where a is a fixed vector, a is its magnitude, and the integration is carried out over the whole space; b) The integral (r^4 + r^2 (r*c) + c^4) delta (r-c)d*tau where V is the volume of a cylinder of radius 6 about the z-axis, the bottom at z = -13, the tope at z = 7, the vector c = 5x + 3y + 2z and c is its magnitude

Physics

 Evaluate the following integrals:a) The integral of (r^2 - 3ar + a^2)delta(r-a)d*tau

where a is a fixed vector, a is its magnitude, and the integration is carried out over the whole space;

b) The integral (r^4 + r^2 (r*c) + c^4) delta (r-c)d*tau

where V is the volume of a cylinder of radius 6 about the z-axis, the bottom at z = -13, the tope at z = 7, the vector c = 5x + 3y + 2z and c is its magnitude.

You have a solid cylinder of radius a inside the co-axial cylindrical shell of radius h. The inner cylinder is uniformly charged with the volumen charge density p. The surface charge density sigma of the outer shell is such that the net charge of the whole system is zero.

a) Find sigma
b) Find the energy per unit length of the system.

You have uniformly charged spherical surface of radius R and total charge q. Consider the sum

where the firs integral is taken over the volume of a sphere of radius a ? R, and the second integral is taken over the surface of this sphere. The vector function E and a scalar function psi are the electric field and potential due to the charge q. The sum gives
a) a fraction of the total electrostatic energy of the charge;
b) total electrostatic energy of the charge;
c) a quantity exceeding the total electrostatic energy of the charge;
d) interaction energy of the charge with all other charges of the Universe
e) none of the above

Two infinite uniform sheets of charge are placed at x = +/- a. The left sheet carries surface charge density sigma, and the right sheet has twice that value.
(a) Find the electric field E(r) in all regions
(b) Plot the graph of the field as a function of x An uncharged spherical conductor of radius R has a cavity of some weird shape. Somewhere within a cavity is a charge -q. Find the electric field E outside the sphere.

pur-new-sol

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case (a):

Since the integration is over the whole space and there is the delta-function, all you have to do is to substitute r = a (vectors) in the part outside the delta-function. Therefore the integral is

I_a = a^2-3a^2+a^2 = -a^2

case(b):

If the integral were over the whole space, you would just substitute r=c as in case (a).
However the integral is over the volume of a cylinder, so you have to check whether c is inside the cylinder or outside. If outside, the integral is 0, if inside you just substitute r=c as in case (a).

Let us find it out.
c_z = 2 and so it is between the bottom (z=-13) and the top (z=7).
The distance of c from the axis is sqrt(5^2+3^2) = sqrt(25+9) = sqrt(34) < 6 (6 is the radius of the cylinder).

Therefore we find that c is inside the cylinder and so we do the same as in case (a), substituting r=c in what is outside the delta-function:

I_b = c^4 + c^2*0 + c^4 = 2c^4

R

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