Two plastic balls, each of radius 17.5 cm, are placed so that there is a 27.2 cm space between them. The charge on one ball is +2.90 nC and the charge on the other is -8.00 nC. What is the magnitude of the electric field at a point directly midway between the two balls?
Question
Two plastic balls, each of radius 17.5 cm, are placed so that there is a 27.2 cm space between them. The charge on one ball is +2.90 nC and the charge on the other is -8.00 nC. What is the magnitude of the electric field at a point directly midway between the two balls?
Solution
To solve this problem, we need to use the formula for the electric field produced by a point charge, which is given by:
E = k * |q| / r²
where:
- E is the electric field,
- k is Coulomb's constant (approximately 8.99 * 10^9 N m²/C²),
- q is the charge, and
- r is the distance from the charge.
First, we need to calculate the electric field produced by each ball at the midpoint. The distance from each ball to the midpoint is half the total distance between the balls, which is 27.2 cm / 2 = 13.6 cm = 0.136 m.
- For the ball with a charge of +2.90 nC (which is 2.90 * 10^-9 C), the electric field at the midpoint is:
E1 = k * |q1| / r1² E1 = 8.99 * 10^9 N m²/C² * 2.90 * 10^-9 C / (0.136 m)² E1 = 1.37 * 10^5 N/C
- For the ball with a charge of -8.00 nC (which is -8.00 * 10^-9 C), the electric field at the midpoint is:
E2 = k * |q2| / r2² E2 = 8.99 * 10^9 N m²/C² * 8.00 * 10^-9 C / (0.136 m)² E2 = 3.77 * 10^5 N/C
Since the charges have opposite signs, the electric fields produced by the two balls at the midpoint are in opposite directions. Therefore, the total electric field at the midpoint is the difference between the two fields:
E_total = |E1 - E2| E_total = |1.37 * 10^5 N/C - 3.77 * 10^5 N/C| E_total = 2.40 * 10^5 N/C
So, the magnitude of the electric field at a point directly midway between the two balls is 2.40 * 10^5 N/C.
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