Question 1: An infinitely long uniformly charged cylinder of radius R is shown.
(a) Using symmetry arguments, sketch the shape of the electric field of the cylinder.
(h) Use Gauss's Law to show that the electric field magnitude within the cylinder is
E(r) = P r, r < R.
2€0
where r is the distance from the centre of the cylinder and p is the volume charge density of the cylinder in units of C/m3.
The electric field outside the cylinder is
E(r) =pR2 P. r > R.
2E0 r
(c) Sketch the electric field magnitude as a function of position r.
A —50 nC point charge of mass 2.0 g is located outside the cylinder at a position rcharne = 3 mm.
(d) If p = 1.0 C/m3 and R = 2.0 mm, calculate the electrostatic force exerted on the point charge by the cylinder.
(e) The point charge is propelled directly away from the cylinder with a velocity of 5.0 m/s. Due to the attractive force between the charge and the cylinder, the point charge decelerates, then stops, and finally moves back towards the cylinder. Calculate the maximum distance that the charge should reach from the centre line of the cylinder.
(f) An experimenter attempts to verify your predictions from part (e), and wishes to achieve an uncertainty of less than 0.1 mm in the maximum distance between point charge and cylinder.
What level of uncertainty would he acceptable in the initial position? You may assume that the uncertainty in all other parameters is negligible.
The following expression may be useful: f xf 1 —dxX;
Question 2: Consider a line of uniform charge Q that is bent into a 1-turn helix of radius R and height h.
Suppose the helix is placed on au insulating surface and held such that the axis of the helix is vertical, as shown in the figure.
(a) Discuss how to calculate the electric field and potential generated by the helix. What are the main differences between calculating the electric field and the electric potential?
(b) Consider a point P that lies on the axis of the spiral where the axis intersects the horizontal surface. Show that the electric potential at point 1-' is:
V = Q In h.+ \/1+12 2
4ricoh R2
(c) To what simpler result does this expression approach in the limit that the height of the spiral goes to zero h 0? In your answer include a mathematical justification, and explain whether this is what you expect based on physical arguments.
Useful mathematical results:
dx
in + Vx2 + a2)
v3.2 + 0,2
ln(ab) = ln(a) + ln(b)
ln(a/b) = ln(a) — ln(b)
ln(1 + x) x if 37 < 1
(1 + 1 + nx if x < 1
Attachment:- Assignment_Q1_Q2.pdf