Question
B. Two charges of 2.0times 10^-6C and -1.0times 10^-6C are placed at a separation of 10 cm. Determine where a third charge should be placed on the line connecting the two charges so that it experiences no net force due to these two charges.A
Solution
4.3
(394 Votes)
Saige
Master · Tutor for 5 years
Answer
Here's how to determine the position of the third charge:**1. Understanding the Problem**We have two charges: a positive charge (q1) and a negative charge (q2). We need to find a point on the line connecting them where a third charge (q3) would experience no net force. This means the force of attraction from the negative charge must balance the force of repulsion from the positive charge.**2. Coulomb's Law**The force between two charges is given by Coulomb's Law:F = k * |q1 * q2| / r^2Where:* F is the force* k is Coulomb's constant (approximately 8.99 x 10^9 N⋅m^2/C^2)* q1 and q2 are the magnitudes of the charges* r is the distance between the charges**3. Setting up the Equation**Let's assume the positive charge (q1) is at the origin (x = 0 cm), and the negative charge (q2) is at x = 10 cm. We'll call the position of the third charge x.* Force due to q1 on q3: F13 = k * |q1 * q3| / x^2* Force due to q2 on q3: F23 = k * |q2 * q3| / (10 - x)^2For the net force on q3 to be zero, F13 must equal F23:k * |q1 * q3| / x^2 = k * |q2 * q3| / (10 - x)^2**4. Solving for x**Notice that k and q3 cancel out:|q1| / x^2 = |q2| / (10 - x)^2Substitute the given values for q1 and q2:(2.0 x 10^-6) / x^2 = (1.0 x 10^-6) / (10 - x)^2Simplify and solve for x:2 / x^2 = 1 / (10 - x)^22(10 - x)^2 = x^2Take the square root of both sides:√2 * (10 - x) = ±xWe have two possible solutions:* **Case 1: Positive root**√2 * (10 - x) = x10√2 - x√2 = x10√2 = x(1 + √2)x = 10√2 / (1 + √2) ≈ 5.86 cm* **Case 2: Negative root**√2 * (10 - x) = -x10√2 - x√2 = -x10√2 = x(√2 - 1)x = 10√2 / (√2 - 1) ≈ 24.14 cm**5. Choosing the Correct Solution**The negative charge (q2) is at x = 10 cm. The third charge must be placed *outside* the two charges for the forces to balance. If it were placed between the two charges, both forces would act in the same direction. Therefore, the correct solution is x ≈ 24.14 cm.**Answer:** The third charge should be placed approximately 24.14 cm from the positive charge along the line connecting the two charges. This position is 14.14 cm to the right of the negative charge.