THE UNIVERSITY OF BRITISH COLUMBIA
 
Science 1 Physics Assignment # 13:
 
ELECTRIC CHARGE & GAUSS' LAW
 
Wed. 24 Jan. 2001 - finish by Wed. 31 Jan.

1.

Units: Moving a charge of one 1 C [one coulomb] through a potential of 1 V [one volt] takes 1 J [one joule] of work (or will produce 1 J of kinetic energy). This handy relationship is a little clumsy in modern physics, which deals with elementary particles whose charge is quantized in units of $e = 1.602 \times 10^{-19}$ C. A convenient energy unit in accelerator physics is therefore the electron volt [eV], defined as the (kinetic) energy gained by an electron (or any other particle with the same size charge) when it is accelerated through a one volt [1 V] electric potential.¹   So . . . how many electron volts in one kilowatt-hour?²

2.

Earth's Extra Electrons: The Earth's atmosphere is a thin electrical insulator which has an excess of positive charge at higher altitude and an excess of negative charge near the Earth's surface, which acts like a spherical conductor. This charge separation gives rise to an approximately constant electric field pointing down through the atmosphere. However, above the atmosphere there is no electric field due to the Earth unless it has a net charge. Assuming that only electrostatic and gravitational forces are involved, calculate how many excess electrons the Earth could hold. (Hint: what will happen to a free electron just outside the atmosphere?)

3.

Triangle of Charges: Three charges are placed at the vertices of an equilateral triangle of side a (as shown) and a fourth charge is placed at the centre of the triangle.
 
 

(a)

Sketch the electric field lines inside and outside (nearby) the triangle.

(b)

Derive an expression for the vector electric field $\vec{E}$ at the centre charge due to the three other charges at the vertices of the triangle.

4.

Gauss' Law for a Sheet of Charge:  Imagine an infinite plane sheet of electric charge with $dQ/dA = \sigma$ units of charge per unit surface area. Using your own words and drawings,³

 

5.

Gauss' Law for a Line of Charge: Imagine an infinite straight line of electric charge with $dQ/d\ell = \lambda$ units of charge per unit length. Using your own words and drawings,

(a)

Using only (i) the SUPERPOSITION PRINCIPLE for the total electric field due to an assembly of electric charges, plus (ii) simple SYMMETRY arguments, deduce the direction of the electric field due to the line of charge.

(b)

Using the preceding result plus the general ideas of GAUSS' LAW and simple geometry (surface area of a cylinder, etc.), deduce the dependence of the magnitude E of the electric field upon r, the perpendicular distance away from the line.

. . . and Tipler (4$^{\rm th}$ Edition)

Ch. 11: problem 78

Ch. 22: problems 50, 51, 54 and 82

Ch. 23: problems 10, 53 and 57

WOOPS!⁴

Ch. 18: problems 33, 44 and 62

Ch. 19: problems 45, 66 and 96

Ch. 20: problem 19