Three charges lie along the x-axis. At x = 2.0m, a charge of 15x10^-6 C is fixed. At the origin, a charge of 30x10^-6 is fixed (like in question #8). Where can a third charge be placed between them so it feels no net force?
To find the position where a third charge can be placed between the two fixed charges so that it experiences no net force, we need to consider the electrostatic forces acting on it.
Let's define the charges:
Q1 = +15x10^-6 C (at x = 2.0 m)
Q2 = +30x10^-6 C (at x = 0)
The magnitude of the electrostatic force between two charges can be calculated using Coulomb's Law:
F = (k * |Q1| * |Q2|) / r^2
where F is the force between the charges, k is Coulomb's constant (9.0x10^9 Nm^2/C^2), |Q1| and |Q2| are the magnitudes of the charges, and r is the distance between them.
Since we want the net force on the third charge to be zero, it means the forces due to Q1 and Q2 must balance each other. This implies that the magnitudes of the forces must be equal:
|F1| = |F2|
Now, let's find the forces individually and set them equal:
|F1| = (k * |Q1| * |Q3|) / r1^2
|F2| = (k * |Q2| * |Q3|) / r2^2
where |Q3| is the magnitude of the third charge.
Since one charge is located at x = 2.0 m and the other is at x = 0, we'll call the distance between the third charge and Q1 r1 and the distance between the third charge and Q2 r2.
To solve for the position of the third charge, we need to solve the equation |F1| = |F2| for r1 and r2.
(k * |Q1| * |Q3|) / r1^2 = (k * |Q2| * |Q3|) / r2^2
|Q1| / r1^2 = |Q2| / r2^2
Now we substitute the given values:
|Q1| = 15x10^-6 C
|Q2| = 30x10^-6 C
Plugging in the values, we have:
(15x10^-6 C) / r1^2 = (30x10^-6 C) / r2^2
To find where the third charge will experience no net force, we need to solve this equation.
Finally, we can rearrange the equation to solve for r1 and r2, the distances between the third charge and Q1 and Q2 respectively. We need to solve the equation:
(15x10^-6 C) / r1^2 = (30x10^-6 C) / r2^2
Cross-multiplying and simplifying:
15x10^-6 C * r2^2 = 30x10^-6 C * r1^2
Now, you can solve this equation to find the values of r1 and r2.
After finding the values of r1 and r2, you can locate the position of the third charge between Q1 and Q2 using the distances r1 and r2.