Part A & B are 0.10m apart. A point charge of +3.010^-9C is placed at A and a point charge of +1.010^-9C is placed at B. X is a point on the straight line through A and B. If a third charge C or charge 1.5*10^-9 is placed at X, where must X be for the resultant force to be zero.

Question

Part A & B are 0.10m apart. A point charge of +3.010^-9C is placed at A and a point charge of +1.010^-9C is placed at B. X is a point on the straight line through A and B. If a third charge C or charge 1.5*10^-9 is placed at X, where must X be for the resultant force to be zero.

Answer 1

To find the position of point X where the resultant force is zero, we need to consider the forces exerted by the two charges at points A and B. We can calculate the electric force between each charge and X using Coulomb's Law. The electric force between two charges is given by:

F = (k * q1 * q2) / r^2

Where F is the force, k is the Coulomb's constant (8.99 * 10^9 N m^2/C^2), q1 and q2 are the charges, and r is the distance between the charges.

Let's start by calculating the force of charge A (+3.0 * 10^-9 C) on point X for a given distance x between A and X. The force is attractive since both charges have the same sign:

Fa = (k * qa * qX) / (x^2)

Next, let's calculate the force of charge B (+1.0 * 10^-9 C) on point X for a distance of (0.10 m - x) between X and B:

Fb = (k * qb * qX) / ((0.10 - x)^2)

To find the position where the resultant force is zero, we need to set the two forces equal to each other:

Fa = Fb

(k * qa * qX) / (x^2) = (k * qb * qX) / ((0.10 - x)^2)

We can simplify this equation by canceling out the Coulomb's constant and the charge qX:

(qa * x^2) = (qb * (0.10 - x)^2)

Now, we can solve this equation for the position x by substituting the given charges:

(3.0 * 10^-9 C) * x^2 = (1.0 * 10^-9 C) * (0.10 - x)^2

Simplifying this equation further, we can expand the square brackets:

3 * x^2 = 1 * (0.10 - x) * (0.10 - x)

3 * x^2 = 0.01 - 0.1x - 0.1x + x^2

2 * x^2 + 0.2x - 0.01 = 0

Now we have a quadratic equation. We can solve it by using the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / (2a)

Applying the values a = 2, b = 0.2, and c = -0.01, we can solve for x:

x = (-0.2 ± √(0.2^2 - 4 * 2 * (-0.01))) / (2 * 2)

Simplifying further:

x = (-0.2 ± √(0.04 + 0.08)) / 4

x = (-0.2 ± √0.12) / 4

x = (-0.2 ± 0.3464) / 4

x = (-0.2 + 0.3464) / 4 OR x = (-0.2 - 0.3464) / 4

x = 0.1464 / 4 OR x = -0.5464 / 4

x = 0.0366 OR x = -0.1366

Since the distance cannot be negative, the only valid solution is x = 0.0366 m.

Therefore, the point X must be located at a distance of 0.0366 meters from point A along the straight line connecting A and B for the resultant force to be zero.

Part A & B are 0.10m apart. A point charge of +3.0*10^-9C is placed at A and a point charge of +1.0*10^-9C is placed at B. X is a point on the straight line through A and B. If a third charge C or charge 1.5*10^-9 is placed at X, where must X be for the resultant force to be zero.

To find the position of point X where the resultant force is zero, we need to consider the forces exerted by the two charges at points A and B. We can calculate the electric force between each charge and X using Coulomb's Law. The electric force between two charges is given by:

Part A & B are 0.10m apart. A point charge of +3.010^-9C is placed at A and a point charge of +1.010^-9C is placed at B. X is a point on the straight line through A and B. If a third charge C or charge 1.5*10^-9 is placed at X, where must X be for the resultant force to be zero.