# 2 q 4 π? 0 parenleftbigg 1 r 1 r d parenrightbigg 3

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2 Q 4 πǫ 0 parenleftbigg 1 R 1 R + d parenrightbigg 3. Q 4 πǫ 0 parenleftbigg 1 L + d 1 R + d parenrightbigg 4. Q 4 πǫ 0 parenleftbigg 1 R + d + L 1 R + d parenrightbigg 5. 2 Q 4 πǫ 0 parenleftbigg 1 L + d 1 R + d parenrightbigg 6. 2 Q 4 πǫ 0 parenleftbigg 1 R + d + L 1 R + d parenrightbigg 7. 2 Q 4 πǫ 0 parenleftbigg 1 L + d 1 R + d parenrightbigg 8. Q 4 πǫ 0 parenleftbigg 1 L + d 1 R + d parenrightbigg 9. Q 4 πǫ 0 parenleftbigg 1 R + d + L 1 R + d parenrightbigg 10. Q 4 πǫ 0 parenleftbigg 1 E 1 R + d parenrightbigg Explanation: Let us label the two spheres as 1 and 2. Sphere 1 is the sphere with a uniformly dis- tributed charge of + Q and Sphere 2 is the sphere with a uniformly distributed charge of Q . Then, by using the principle of superpo- sition, we have V A = V 1 ,A + V 2 ,A V A = 1 4 πǫ 0 Q R + d + 1 4 πǫ 0 ( Q ) R + d + L Similarly, we may write V B = V 1 ,B + V 2 ,B V B = 1 4 πǫ 0 Q R + d + L + 1 4 πǫ 0 ( Q ) R + d Combining them, we obtain the potential dif- ference, Δ V = V B V A . Δ V = 1 4 πǫ 0 parenleftbigg Q R + d + L Q R + d parenrightbigg 1 4 πǫ 0 parenleftbigg Q R + d Q R + d + L parenrightbigg Δ V = 1 4 πǫ 0 parenleftbigg 2 Q R + d + L 2 Q R + d parenrightbigg Δ V = 2 Q 4 πǫ 0 parenleftbigg 1 R + d + L 1 R + d parenrightbigg Thus, the potential difference is V B V A = 2 Q 4 πǫ 0 parenleftbigg 1 R + d + L 1 R + d parenrightbigg 013 10.0points Given three parallel conducting plates which are aligned perpendicular to the x-axis. They are labeled, from left to right as plate 1, 2 and 3 respectively. The corresponding plate charges are Q 1 = 2 q , Q 2 = q and Q 3 = 3 q . The width of the gap between 1 and 2 is d which is the same as the width across 2 and 3. Determine the Δ V = V 1 V 3 . 1. 11( q/A ) d 2 ǫ 0 2. 7( q/A ) d 2 ǫ 0 3. 2( q/A ) d ǫ 0 4. 5( q/A ) d ǫ 0 correct 5. 4( q/A ) d ǫ 0 6. 3( q/A ) d ǫ 0 7. 9( q/A ) d 2 ǫ 0 8. 5( q/A ) d 2 ǫ 0 9. 6( q/A ) d ǫ 0 Explanation: One may regard the 3-plate system as a composite system which involves two capaci- tor systems with the 12-capacitor followed by the 23-capacitor. The 12-capacitor has charges Q 1 and Q 2 + Q 3 , i.e charges of +2 q and 2 q respectively.
sarceno (dea457) – Ch17-h2 – yao – (57465) 8 The 23-capacitor has charges Q 1 + Q 2 and Q 3 , i.e charges of +3 q and 3 q respectively. Note that the charges on plate 1 and 3 are 2 q , 2 q + 3 q = q and 3 q which are in agreement with the given charges. The potential difference is V 1 V 3 = E gap, 12 d + E gap, 23 d V 1 V 3 = 2( q/A ) d ǫ 0 + 3( q/A ) d ǫ 0 V 1 V 3 = 5( q/A ) d ǫ 0 Digression: Notice that E is pointing to the right. This implies that the ”potential hill” has a downward slope to the right. Going from plate 3 to plate 1, corresponds to moving to the left, which is climbing the potential hill. This implies V 1 V 3 > 0. 014(part1of2)5.0points A very long glass rod of length 2 R carries a uniformly distributed charge + q as shown in the figure. A very large plastic disk of radius R , carrying a uniformly distributed charge Q is located at a distance d from the rod, where d << R . Determine the potential difference Δ V rod = V B V A due to the long
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