1. The diagram below shows a voltage-bias transistor circuit. In the textbook by Floyd, a number of approximations are introduced to "solve" the circuit. For instance, I_E is replaced by I_C and we "look in" to the transistor to determine V_B, the base to ground potential. The problem is that the simplicity of the resulting equations is outweighed by the confusion this approach generates.
   

   For the circuit below, use Kirchhoff's Rules and I_C = β_DCI_B only, to
   (a) find I_B in terms of β_DC, V_CC, V_BE, and the resistances R₁, R₂, R_C, and R_E, and;
   (b) find V_CE in terms of β_DC, I_B, V_BE, and the resistances R₁, R₂, R_C, and R_E, and;
   (c) find V_B in terms of β_DC, V_CC, V_BE, and the resistances R₁, R₂, R_C, and R_E.

   If V_CE(sat) = 0.35 V, V_BE = 0.65 V, β_DC = 170, R₁ = 20 kΩ, R₂ = 5 kΩ, R_C = 3 kΩ, R_E = 500 Ω, determine the following using theyour equations.

   (d) Find the Q-point if V_CC = 8.00 V.
   (e) Determine the value of V_B if V_CC = 8.00 V.
   (f) Determine the values of I_B, I_C, and V_CC at saturation.
   (g) Determine the value of V_CE and V_CC at cutoff.
   (h) Also draw the DC load line on a graph of I_C versus V_CE. Indicate the Q-point.
   

   

2. A flat disk of radius R and infinitesimal thickness has a uniform surface charge density Σ.
   (a) Find the electrostatic potential at the centre of the disk by using direct integration.
   (b) Find the electrostatic potential at the edge of the disk, at P, using direct integration. Hint - the hard part of this problem is determining the limits of integration. Show, for fixed angle θ, that r has a maximum value of 2Rcos(θ). See the diagram below.
   (c) Using your above answers, explain why the disk is not a conductor in equilibrium.
   
   

3. Below are some electric fields. Determine which are possible electrostatic fields. For those which meet the criterion, determine an electrostatic potential which could produce them, the charge density which produced them, and the total charge within a cube of unit side length centered at (0,0,0).
   (i) E_x = x/ε₀, E_y = y/ε₀, and E_z = z/ε₀;
   (ii) E_x = (x²+yz)/ε₀, E_y = (y²+xz)/ε₀, and E_z = (z²+xy)/ε₀;
   (iii) E_x = 2xy/ε₀, E_y = (2z+3)/ε₀, and E_z = (5-2yz)/ε₀;
   (iv) E_x = (y³-z³)/ε₀, E_y = (z³-x³)/ε₀, and E_z = (x³-y³)/ε₀.
   

4. The diagram below shows the cross-section of a spherical distribution of charge. It consists of a sphere of radius R/2 which has a charge 2Q/3 uniformly distributed throughout. Around the sphere is a thick spherical shell of thickness R/6. The shell has a density ρ(r) = -9Q/(7πR²r). Next is a gap. Then there is a thin spherical shell with a surface charge density Σ = Q/12πR².
   (a) Determine the electric field as a function of the distance from the centre of the sphere.
   (b) Sketch the electric field.
   
   

5. The Method of Images is a powerful technique for finding charge distributions on conducting surfaces. The diagram below shows a long conducting cylinder of radius R and a long straight wire with linear charge density λ located a distance b from the centre of the cylinder. The Method of Images suggests that the cylinder can be replaced with an image wire a distance a from the centre of the cylinder with charge density λ´.
   (a) Confirm (no need to derive the result) that λ´ = -λ and a = R²/b, will cause a cylindrical surface at R to be at a constant potential (i.e. an equipotential) for all angles .
   (b) What is the potential of the cylinder?
   (c) Confirm that E_net due to the wire and its image is purely radial then obtain an expression for Σ = ε₀E_net in terms of λ, R, b, and θ.
   The potential due to a wire is V_wire = -λ[ln(r/C)]/2pe₀ where C is a constant. The electric field due to a wire is E_wire = λ/2pe₀r and is radial. Hint - in solving the problem, use the Law of Cosines and the Law of Sines to find r´ in terms of r, and sin(α), cos(α), sin(α+β), and cos(α+β) in terms of r, R, b, and θ.
   
   
   

6. Determine the magnetic field at point P using the Law of Biot-Savart. The wire is in one plane and the wires are very long (semi-infinite is the phrase used).
   
   

Questions? mike.coombes@kwantlen.ca