Question

Problem 4. A positively charged wire is bent into a semicircle of radius R , as shown in the figure below. The total charge on the semicircle is Q . The charge per unit length along the semicircle is uniform.

a) What is the Electric field at the origin?

b) If a particle with a charge q is placed at the origin, what is the total force on the particle? Show all your work including setting up and integrating any necessary integrals.

c) What is the potential at the origin?

d) If a particle with a charge q is placed at the origin, what is the potential energy of the particle?

Problem 5. The gap between the electrodes in an automobile engine’s
spark plug is 0.060 cm. To produce an electric spark in a gasoline-air mixture, an electric field of 3.0 × 10⁶ V/m must be achieved.

(a)On starting a car, what minimum voltage must be supplied by the
ignition circuit? Assume the field is uniform.

(b) The small size of the gap between the electrodes is inconvenient
because it can get blocked easily, and special tools are needed to
measure it. Why don’t they design spark plugs with a wider gap?

Problem 6. Astronomers believe that the mass distribution (mass per
unit volume) of some galaxies may be approximated, in spherical
coordinates, by ρ = ae^−br, for 0 ≤ r ≤ ∞, where ρ is the density.
Find the total mass.

Answer 1

a) Consider the figure below: we have: dĒ = dE(-cosei - sinoſ) Where: kdQ dE = R2 In this case: dQ = idl = 2Rde Then: κλ dĒ = *(-cosodeî - sinedeſ) Integrating this expression 0 <o< Ē = m ( -cose doi – ("sino dej) _ 2k1 E=RÎ The uniform density: 2e AR Then: E _2kQ TER b) The force is: F = qĒ - 2kQq TR2) c) The electric potential at the origin: kdQ dv = Ã v Integrating this expression: kokQ dQ= d) The electric potential energy is: U = qv = kg