Download ph504-1213-ass - University of Kent

PH504/12/13a
UNIVERSITY OF KENT
SCHOOL OF PHYSICAL SCIENCES
2012/13
Assignment 1: Analysis
1. Calculate the gradient () of the following functions (assume Cartesian co-ordinates)
(a) f = 3x2+2y2+z2,
(b) f = x2y2z2,
(c) f = exyz,
(d) f = 4 sin(xyz) .
(e) ex sin(xyz) .
2. A vector is given by -iy+jx in Cartesian coordinates with unit vectors (i , j).
(a) Calculate the line integral of this vector anti-clockwise around the sides of the square
defined by the co-ordinates (0,0), (a,0), (a,a), (0,a).
(b) Show that the full result is equal to the curl of the vector multiplied by the area of the
square (this is Stoke's theorem).
3. (a) Suppose that a = 3i + 5j - 2k and b = 2i - 2j - 2k
where i, j and k are orthogonal unit vectors. Calculate the dot product and vector product.
What is the angle between the vectors? 90 degrees
(b) Suppose that
c = 2i - 3j + k and d = 4i + j - 3k
Calculate the angle between the vectors in radians.
2
Assignment 2: Charge
1. A line charge density is given by(x) = 3x3 C m-1. Calculate the total charge
contained between x = 0 and x = 1.
2. A surface charge density (x,y) is given by (x,y)=3x2+4y2-xy Cm-2. Calculate the
total charge contained within the area bounded by x=0+a, y=0+a.
Assignment 3: Electric field
1. State how the electric field is related to the electrostatic potential.
2.
In some region of space, the electrostatic potential takes the following functional form
in Cartesian coordinates x, y, and z: V(x,y,z) = 1/x2 + 1/(xy), where the potential is
measured in volts and the distances in meters. Find the electric field, E, at the point x
= 2 m, y = 2 m, z = 2m. What is the value of curl E?
3. By applying Gauss' law to a charged conducting sphere of radius r and
charge Q, the magnitude of the electric field outside the sphere is found to be
E = + Q /(40r2). Detemine the electric potential and, hence, derive a formula
for the self-capacitance of the sphere.
4. Calculate the self-capacitance of the Earth in units of nF.
3
5. A conducting cylinder of radius Rc carries a uniform surface charge of +per unit
length. Determine the electric field strength inside and outside the cylinder.
6. Air ionises for E-fields above 30 kV cm-1. Calculate the maximum charge per unit
length that can be placed on a conducting cylinder of radius 2 cm without the
surrounding air ionising.
Assignment 4: Capacitance
1. By consideration of the electric charge and potential, determine the total capacitance of a
system of N capacitors of capacitance C1, C2, …….CN, placed in parallel.
3. Determine the total capacitance of a system of N capacitors of capacitance C1, C2,
…….CN, placed in series.
4.
3. A group of identical capacitors is connected first in series and then in parallel. The
combined capacitance in parallel is 529 times larger than for the series connection. Determine
the number of capacitors.
4. A parallel plate capacitor has plates of area 10 cm2 separated by 5mm.
a). Calculate the capacitance and the potential difference between the plates for a
charge of 5x10-11 C.
b). Determine the energy stored by the capacitor and the magnitude of the E-field
between the plates?
c.) An electron is released from rest from one of the plates and is accelerated towards the
second plate. Find the speed of the electron when it hits the second plate.
turn over
4
5. A line charge having a constant charge density of +1 C m-1 extends along the y-axis
from -∞ to +∞. Calculate the E-field at a distance of 1 millimetre along the positive x-axis.
Take = 8.85x10-12 C2 m-2 N-1 .
[20 marks]
6. Consider an inner conducting metal sphere with total charge -q of radius a and a thin outer
conducting metal shell with total charge +2q and radius b (of negligible thickness). Write
down expressions for the electric field and electric potential in the three regions: (i) within the
sphere (ii) between the sphere and the shell and (iii) external to the shell. Sketch graphs of the
radial distributions of the electric field and the electric potential (please label your axes).
b
a
-q
+2q