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.
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”
To find the point between the two charges where a charge of +3.1 × 10^-9 C should be placed such that the net electric force on it is zero, we can apply the principle of superposition and use Coulomb's Law.
The electric force between two charged particles is given by Coulomb's Law:
F = (k * q1 * q2) / r^2
Where:
F is the force between the charges,
k is the Coulomb's constant (k ≈ 9 × 10^9 N m^2/C^2),
q1 and q2 are the magnitudes of the charges,
and r is the distance between the charges.
In this case, we have two charges and we want to find the point where the net electric force is zero. Let's start by finding the electric forces due to each of the charges at a general point (x, 0). We'll call the distance from the origin to this point as "d1" and the distance from the point to the charge at x = 2.5 m as "d2".
1. Electric force due to the charge of +2.4 × 10^-9 C:
F1 = (k * q1 * q) / r1^2
Here, q1 = +2.4 × 10^-9 C, q = +3.1 × 10^-9 C (the charge we want to place), and r1 = d1.
2. Electric force due to the charge of +4.2 × 10^-9 C:
F2 = (k * q2 * q) / r2^2
Here, q2 = +4.2 × 10^-9 C, q = +3.1 × 10^-9 C (the charge we want to place), and r2 = d2.
Since we want the net electric force to be zero, the two forces must be equal in magnitude, but opposite in direction:
|F1| = |F2|
F1 = -F2
Let's equate the two forces:
(k * q1 * q) / r1^2 = -(k * q2 * q) / r2^2
Now, substitute the values:
k ≈ 9 × 10^9 N m^2/C^2
q1 = +2.4 × 10^-9 C
q2 = +4.2 × 10^-9 C
q = +3.1 × 10^-9 C
r1 = d1
r2 = |x - 2.5|
(k * q1 * q) / r1^2 = -(k * q2 * q) / r2^2
(9 × 10^9 * 2.4 × 10^-9 * 3.1 × 10^-9) / d1^2 = -(9 × 10^9 * 4.2 × 10^-9 * 3.1 × 10^-9) / (|x - 2.5|)^2
Simplifying and cross multiplying:
2.484 / d1^2 = -13.608 / (x - 2.5)^2
Now, we can solve this equation to find the value of x.
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 principle of superposition.
The electric force is given by Coulomb's law, which states that the magnitude of the force between two point charges is given by:
F = k * (|q1| * |q2|) / r^2
where F is the force, k is Coulomb's constant (8.99 × 10^9 N m^2 / C^2), |q1| and |q2| are the magnitudes of the charges, and r is the distance between the charges.
Let's calculate the forces exerted on the charge of +3.1 × 10−9 C by the charges at the origin and at x = 2.5 m:
1. Calculate the force exerted by the charge at the origin:
F1 = k * (|q1| * |q3|) / r1^2
where |q3| is the magnitude of the charge to be placed between the charges and r1 is the distance between the charge at the origin and the point to be found.
2. Calculate the force exerted by the charge at x = 2.5 m:
F2 = k * (|q2| * |q3|) / r2^2
where |q2| is the magnitude of the charge at x = 2.5 m and r2 is the distance between the charge at x = 2.5 m and the point to be found.
Since we want the net electric force on the charge of +3.1 × 10−9 C to be zero, we need to find the point where the magnitudes of the forces exerted by the two charges are equal:
|F1| = |F2|
Now, we can solve for the point by setting up and solving the equation:
k * (|q1| * |q3|) / r1^2 = k * (|q2| * |q3|) / r2^2
Simplifying the equation:
|q1| / r1^2 = |q2| / r2^2
Substituting the given values:
(2.4 × 10−9 C) / r1^2 = (4.2 × 10−9 C) / r2^2
Solving for the ratio of r1^2 / r2^2:
r1^2 / r2^2 = (4.2 × 10−9 C) / (2.4 × 10−9 C)
r1^2 / r2^2 = 1.75
Taking the square root of both sides:
r1 / r2 = sqrt(1.75)
Now, we can find the value of the ratio r1 / r2:
r1 / r2 ≈ 1.3229
To find the coordinates of the point, we can use the fact that the distance between the charge at the origin and the point to be found, r1, is equal to the distance between the charge at x = 2.5 m and the point to be found, r2, multiplied by the ratio r1 / r2:
r1 = 1.3229 * r2
Since the charge at x = 2.5 m is located at x = 2.5 m, we can substitute the value of r2:
1.3229 * r2 = 2.5 m
Now we solve for r2:
r2 = 2.5 m / 1.3229
r2 ≈ 1.889 m
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 approximately at x = 1.889 m.