Question 1. Two small metal spheres carry two charges q₁ = 0.5 mC and q₂ = 0.3 mC. What is the distance between the two spheres if they experience a force of 850 mN? - Assume that the objects can be approximated by point charges because their size is smaller than their separation. Hint: To compute the correct answer, units must be matched.

Question 2. Show that the amount of work required to assemble four identical charges (from infinity (essentially from very far)) of magnitude qc at the corners of a rectangle of sides d and 2d to a good approximation is 3.894q_c²/(4π∈₀d).

Question 3. Three point charges A, B, and C are arranged in the corners of a square as shown with d₁ = 2 m and d₂ = 3 m. Charge A is 8 mC, charge C is 8 mC, and charge B is -3 mC.

(a) What are the magnitude and direction of the net electrostatic force on charge B due to charges A and C?

(b) What is the direction and magnitude of the electric field at point P?

(c) Find how much work has to be done to move a charge q = 5 mC from infinity to the point P.

Question 4. (a) Use your own words to express Gauss' law for electrostatics, and present an equation that describes this law. Carefully identify each term in the equation including flux.

(b) A hollow, thin, conducting sphere of radius R carries a charge Q. Use Gauss's law to find an expression for the electric field at any radius r inside and outside the sphere. Use diagrams to illustrate your answer.

(c) A thin-walled metal spherical shell of radius a has a charge qa. Concentric with it is a thin-walled metal spherical shell of radius b (where b > a) that has a charge qb. Use Gauss's law to find the electric field at points a distance r from the common center, where

i. r < a,
ii. a < r < b, and
iii. r > b.
iv. Discuss the criterion you would use to determine how the charges are distributed on the inner and outer walls of these shells.

Question 5. How much electric potential energy does a proton gain as it increases its potential by 58 kV?

Question 6. Find the equivalent capacitance of the following circuit across terminals a and b, the voltage across each capacitor and the charge on each capacitor. Assume that C₁ = 2µF, C₂ = 5µF, C₃ = 4µF, C₄ = 6µF, C₅ = 3µF and V0 = 40 V.

Question 7.

(a) Define voltage and electric field, and explain what 40 V and E = 20kˆ N/C mean?

(b) Define capacitance, and explain how capacitors work. Write down an expression that captures the relationship between charge and potential difference between the plates of a capacitor.

(c) Derive an expression for the capacitance of the parallel plate capacitor. Assume that the capacitor carries a +Q charge on one plate and a -Q charge on another plate. Hint: Use Gauss's Law.

(d) Explain in detail why a dielectric material increases the capacitance of a capacitor when placed between the plates?

Question 8.  Four charges are placed around the boundary of an isosceles triangle as shown. Assume that q1 = 2 µC, q2 = -8 µC, q3 = -1 µC, q4 = 2 µC, d1 = 2 cm, and d2 = 2 cm.

(a) Calculate the electric potential at the point "P" .

(b) Calculate the electric field vector at the same point "P" due to the four charges. Room for working:

Question 9. Calculate the current through each component, and the voltage across each com- ponent in the given circuit. Assume that R1 = 3 ?, R2 = 6 ?, R3 = 4 ?, R4 = 8 ?, R5 = 10 ?, R6 = 2 ?, and V = 60 V.

Question 10. The emfs and resistances in the given circuit have the following values: V1 = 20 V, V2 = 15 V, V3 = 34 V, r1 = 10 ?, r2 = 5 ?, r3 = 8 ?, and R = 6 ?.

(a) Find the currents I1 and I2 in this circuit.

(b) Hence, find the potential difference across resistors r1, r2, r3, and R.

(c) What is the potential difference between the terminals of the batteries? That is find Vab, Vac, and Vcd.

Question 11. Two long straight wires are parallel to each other and spaced 0.8 m apart. One of them carry current of 25 A the other 15 A in (a) opposite direction and (b) same direction. Find the magnitude and direction of forces per unit length between these wires in both of these cases.

Question 12. State the Lorentz Force Law, define all the terms, and use it to determine the initial direction of the deflection of charged particles as they enter the magnetic field moving with velocity v shown below.

Question 13.

(a) Describe Ampere's law in words and also with an equation. Explain each term in your equation.

(b) Use Ampere's law to find the magnetic field B^→ a direct current I_c. at a distance r from a long wire that carries

(c) Assuming that we have a direct current and all the current flows along the surface of the wire, find the magnetic field B^→ marks for this question. inside the wire.

(d) Assuming that we have a direct current and all the current flows along the surface of the wire, find the magnetic field B^→ marks for this question. outside the wire.

Question 14.

(a) Describe Faraday's law of induction in words and also with an equation. Explain each term in your equation.

(b) A conducting metal rod ab is in contact with metal rails as shown on the figure below. A uniform magnetic field B^→_in = -12 T (k^{^}) is applied everywhere perpendicularly into the page as shown. The distance between contact points a and b is d = 8 cm.

i. Find the magnitude of the emf induced in the rod as it moves to the right at a velocity 4 m/s to the right.

ii. Explain which end of the rod is at the higher potential.

iii. Explain which direction the current is flowing in the rod.

iv. Draw a clear diagram to show the vector quantities involved.

(c) If the resistance of the circuit around the loop abcd is 60 ? find the current flowing in this circuit, and the magnitude and direction of the force required to keep the rod moving.

(d) Find the energy dissipated through heat as a result of the magnetic force for the time interval ?t = 20 s.

(e) Repeat part (b) with v^→ pointing in the opposite direction (i.e. v^→ is 4 m/s to the left).

Question 15. An initially stationary positive ion of charge q = 1.60 × 10^-19 C and mass m = 3.60 × 10^-25 kg is accelerated through a potential difference V₀ = 5.00 × 10^-3 T before it enters a uniform magnetic field that points out of page with magnitude |B_out| = 8.00 × 10^-3 T as shown.

(a) Use conservation of energy to find the speed of the ion as it enters the uniform magnetic field.

(b) Hence, find the radius r of the ion semicircular path using Lorentz Force law.

Question 16. Find the equivalent resistance between terminals a and b of the circuit shown below for R₁ = 10?, R₂ = 5?, R₃ = 4?, R₄ = 2?, R₅ = 8?, R₆ = 3?, and R₇ = 6?. Give the answer to two decimal places.