3. (10 points) Suppose two infinite planes of charge are parallel to each other (Figure 3), one positively charged and the other negatively charged. The positively and negatively charged planes have charge densities of \( \sigma{L} \) and \( -\sigma{R} \), respectively. What are the directions and magnitudes of the electric fields at point P1 (to the left of the positively charged plane, 3 points), P3 (to the right of the negatively charged plane, 3 points) and P2 (between the two planes, 4 points).

4. (10 points) As shown in Figure 4, a charged particle with \( q \) Coulomb and \( m \) kg passes through a region with the first magnetic field \( B{in} \) and the electric field \( E \). It is known that the charged particle can pass through the region without deflection when the charged particle moves in a specific velocity. After entering the second magnetic field \( B{0,in} \), the charged particle moves in a semicircle of radius \( r \) before striking a detector at \( P \). Please find \( r \) in terms of \( m \), \( B{in} \), \( E \), \( B{0,in} \), and \( q \).

5. (10 points) A long solenoid of radius \( R \) has \( n \) turns of wire per unit length and carries a time-varying current that varies sinusoidally as \( I = I{\max} \cos \omega t \), where \( I{\max} \) is the maximum current and \( \omega \) is the angular frequency of the alternating current source (Figure 5). Please find the magnitudes of the electric field at the points on the dash circle with radius \( r \) for the cases \( r < R \) (5 points) and \( r > R \) (5 points). Assume the magnitude of the magnetic field inside the solenoid is uniform.

6. (10 points) The conducting bar illustrated in Figure 6 moves on two frictionless, parallel rails in the presence of a uniform magnetic field directed into the page. The bar has mass \( m \), and its lengthis $\ell$. The bar is given an initial velocity $\vec{v}_t$ to the right and is released at $t = 0$. Please find the magnitude of the velocity of the conducting bar as a function of time.

7. (10 points) Figure 7 shows an RL circuit ready to switch from node a to node b. Assume before switching, the current through the inductor has reached the maximum current (i.e. \( V / R \)), please prove that the current \( I(t) \) after switching to node b is \( I(t) = (V / R)e^{-t / (L / R)} \).

8. (10 points) Two infinitely long, parallel wires are lying on the ground a distance \( a \) meter apart as shown in Figure 8. A third wire, of length \( L \) meter and mass \( m \) kg, carries a current of \( I{1} \) Ampere and is levitated above the first two wires, at a horizontal position midway between them. The infinitely long wires carry equal currents \( I{2} \) in the same direction, but in the direction opposite that in the levitated wire. What current \( (I_{2}) \) must the infinitely long wires carry so that the three wires form an equilateral triangle?

9. (10 points) The electric potential at the position of a point \( P \) in three dimensions be given by the vector \( r = (x, y, z) \) in Cartesian Coordinates is \( (x^2 + y^2 + z^2)^{-0.5} \). Please find the electric field at the point \( (1, 2, 3) \).

1. 電力系統某點發生短路故障，往系統看之正序、負序、零序阻抗分別為 Z1、Z2、Z0，假設 Z1＝Z2、Z0＝2Z1＝2Z2，故障阻抗 ZF＝0，則兩線短路時正序故障電流大小，為三相短路時正序故障電流大小之_______倍。

2. 承上題，兩線接地時零序故障電流大小，為單線接地時零序故障電流大小之_______倍。

3. 承上題，兩線接地時，正序故障電流大小為負序故障電流大小之_______倍。

4. 一電力系統有甲、乙、丙三部發電機併聯運轉，發電量分別為 P1(MW)、P2(MW)、P3(MW)， 總負載為 PL(MW)，忽略發電限制與不計輸電損失，發電機的運轉成本分別為：C1＝550 ＋6.36 P1 ＋0.0032 2 P1 元／hour、C2＝500＋6.60 P2 ＋0.0048 2 P2 元／hour、C3＝450＋6.96 P3＋0.0072 2 P3 元／hour。如果三部發電機經濟調度時，遞增運轉成本為 10.2 元／MW hour，則甲部發電機的發電量為_______MW。

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