1)In the circuit shown below, a resistor with resistance R is connected in series with a capacitor with capacitance C, a switch S and a battery of e.m.f. ε and internal resistance r. Initially

Revision Quiz 8 on CW 18,19,20 (L34 – L41)

(31 marks in ≈ 30 minutes)

1) In the circuit shown below, a resistor with resistance R is connected in series with a capacitor with capacitance C, a switch S and a battery of e.m.f. ε and internal resistance r. Initially the switch S is open and the capacitor is uncharged. The values of the components are as follows:

ε = 6.00, V, r = 0.300 kΩ, R = 2.00 kΩ, C = 50.0 μF.

a) Calculate the current I ₀ flowing in the circuit at the time t = 0 when the switch is closed.

b) (i) Use Kirchhoff’s second law to write the first order differential equation describing the charging of the capacitor.

(ii) By using the method of separation of variables, solve this differential equation to find an expression for the charge q as a function of time t.

(7 marks)

2) a) An AC circuit is made of a resistor R that is connected to an AC source of instantaneous voltage D v =D V _max sin(ωt).Show that the average power delivered by the source to the resistor over one period is given by:

b) A solenoid of length l has N turns and cross-sectional area A. A variable current i creates a gradient d i /d t in the solenoid. Use Faraday’s law and Lenz’s law of self-induction to show that the expression for the inductance L of the solenoid is:

where m ₀ is the diamagnetic permeability of free space. (10 marks)

3) A mass spectrometer is used to separate two singly and positively charged helium isotopes, , and The ions first pass through a velocity selector before entering a region where only a magnetic field exists. The magnitudes of the electric and magnetic fields are 100V cm-1, and 0.2T, respectively. Their directions are shown in the figure below. The atomic masses of , and are 3.016 u and 4.003 u, respectively.

a) Calculate the speed at which the ions can pass through the velocity selector.

Ions then enter a region where only the magnetic field is present (represented by the dotted area in the above figure).

b) What is the direction of the force acting on the ions as they enter this region?

c) After the ions have travelled through semi-circles, they hit a screen. What is the distance between them on the screen? (6 marks)

4) In the diagram shown below a single loop (N =1) of radius r = 2.50m moves towards a uniform magnetic field B = 2.50 T with a constant speed v. It takes the loop a time D t = 3.14 s to be totally placed inside the magnetic field. The loop has a resistance R = 3.50 Ω.

a) Calculate the constant speed v.

b) By using Faraday’s law or otherwise, calculate the average induced emf between points C and D.

c) By using the conservation of mechanical power find the magnitude F of the average mechanical force that is required to maintain the constant speed v of the loop as it enters the magnetic field B. (8 marks)

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