Electromagnetic Theory
Constants:
- k = 9.10 Nm/C
- e = 1.6 * 10^-19 C
- μ₀ = 8.85 * 10^-12 C^2/Nm^2
- μ = 4π * 10^-7 TmA
Q1: Let A = -a + a + a, B = -a + a + a, C = -a + 2a. Find A·B, A×B, |A×B|, and C·(A×B).
mâ‚ = 9.11 * 10^-31 kg
mâ‚‚ = 1.67 * 10^-27 kg
Q2:
(a) Show that A·B + A×B - A·B = 0.
(b) Show that A×A = 0 and A·(A×A) = 0.
Q3: If A = -4a - 6a + a and B = -2a + 5a, find A·B + 2|B| and a unit vector perpendicular to both A and B.
Q4: Let A = 20a + 15a - 10a and B = a + a. Find A·B, B·B, and the component of A along B.
Q5: Let P = 2a - 4a + a and Q = a + 2a. Find R, which has magnitude 4 and is perpendicular to both P and Q.
Q6:
(a) Prove that P = cosθa + sinθa and Q = cosφa + sinφa are unit vectors in the xy-plane, respectively, making angles θ and φ with the x-axis.
(b) By means of dot product, obtain the formula for cos(θ - φ). By similarly formulating P and Q, obtain the formula for cos(θ + φ).
(c) If θ is the angle between P and Q, find P - Q in terms of θ.