A charge of +2.4 × 10−9 C is placed at the
origin, and another charge of +4.2 × 10−9 C
is placed at x = 2.5 m.
Find the point (coordinate) between these
two charges where a charge of +3.1 × 10−9 C
should be placed so that the net electric
force on it is zero.
To find the point between the two charges where a charge of +3.1 × 10^(-9) C should be placed so that the net electric force on it is zero, we can use the concept of Coulomb's Law.
Coulomb's Law states that the electric force between two charges is given by the formula:
F = (k * |q1 * q2|) / r^2
where:
F is the force between the charges,
k is the electrostatic constant (k = 8.99 * 10^9 Nm^2/C^2),
q1 and q2 are the magnitudes of the charges,
r is the distance between the charges.
Given:
q1 = +2.4 × 10^(-9) C (charge at the origin)
q2 = +4.2 × 10^(-9) C (charge at x = 2.5 m)
q3 = +3.1 × 10^(-9) C (charge at unknown position)
We want to find the distance between q1 and q3.
Let's calculate the force between q1 and q3:
F1 = (k * |q1 * q3|) / r1^2
And the force between q2 and q3:
F2 = (k * |q2 * q3|) / r2^2
Since we want the net electric force on q3 to be zero, F1 and F2 should be equal in magnitude and opposite in direction:
F1 = F2
(k * |q1 * q3|) / r1^2 = (k * |q2 * q3|) / r2^2
We can rearrange the equation to find the coordinate of q3:
r2^2 / r1^2 = |q1 / q2| * |q3|
Substituting the given values:
(2.5^2) / (0^2) = (2.4 × 10^(-9) C / 4.2 × 10^(-9) C) * (3.1 × 10^(-9) C)
Simplifying:
6.25 = (2.4 / 4.2) * 3.1
6.25 = 1.371428 * 3.1
6.25 = 4.251428
To find the coordinate, we need to take the square root of both sides:
r2 / r1 = √(6.25)
r2 / r1 = 2.5
Therefore, the point (coordinate) between the two charges where a charge of +3.1 × 10^(-9) C should be placed so that the net electric force on it is zero is at a distance of 2.5 meters from the origin (charge q1).
To find the point (coordinate) between the two charges where a charge of +3.1 × 10−9 C should be placed so that the net electric force on it is zero, we can use the principle of superposition.
The electric force between the two charges is given by Coulomb's Law:
F = k * (q1 * q2) / r^2
Where:
F is the electric force between the charges,
k is the electrostatic constant (k = 9 x 10^9 Nm^2/C^2),
q1 and q2 are the magnitudes of the charges, and
r is the distance between the charges.
We can divide the problem into two parts:
1. The net electric force between the +2.4 × 10−9 C charge at the origin and the +3.1 × 10−9 C charge at an unknown coordinate.
2. The net electric force between the +4.2 × 10−9 C charge at x=2.5m and the +3.1 × 10−9 C charge at the same unknown coordinate.
For the first part:
F1 = k * ((2.4 × 10−9 C) * (3.1 × 10−9 C)) / r1^2
For the second part:
F2 = k * ((4.2 × 10−9 C) * (3.1 × 10−9 C)) / r2^2
Since we want the net electric force on the +3.1 × 10−9 C charge to be zero, the magnitudes of F1 and F2 should be equal.
Thus,
F1 = F2
k * ((2.4 × 10−9 C) * (3.1 × 10−9 C)) / r1^2 = k * ((4.2 × 10−9 C) * (3.1 × 10−9 C)) / r2^2
Simplifying:
(2.4 × 3.1) * 10^-18 / r1^2 = (4.2 × 3.1) * 10^-18 / r2^2
Dividing both sides by (3.1 × 10^-18) and cross-multiplying:
r2^2 = (4.2 × r1^2) / 2.4
Taking square roots of both sides:
r2 = sqrt((4.2 × r1^2) / 2.4)
Since r1 + r2 = 2.5m, we can substitute this equation into the above expression:
r1 + sqrt((4.2 × r1^2) / 2.4) = 2.5
Now, we can solve this equation for r1 using numerical methods like iteration or trial and error. We can start by making an initial guess for r1 and refine it until we get an accurate answer.
Once we have the value of r1, we can substitute it into r2 = 2.5 - r1 to find the value of r2.
This will give us the coordinates (r1, r2) between the two charges where the net electric force on a +3.1 × 10−9 C charge will be zero.
Since q1 and q2 are positive charges, q3 in its equilibrium position has to be between them at the distance “x” from the charge q1.
F1=F2.
Use Coulomb’s Law
F1= k•q1•q3/x²,
F2 = k •q2•q3/(r-x)²,
k•q1•q3/x²= k •q2•q3/(r-x)²,
q1 /x²= q2 /(r-x)².
Solve for “x”