Sphere A, with a charge of +64 C, is positioned at the origin. A second sphere, B, with a charge of -16 C, is placed at +1.00 on the x-axis. Where must a third sphere, C, of charge +12 C, be placed so that there is no net force on it?
F=0 if E = 0
The point is located to the right from the second charge at the distance ‘x’ =>
kq₁/(X+x)² =kq₂/x²
X=1
64/(1+x)² = 16/ x²,
3x²-2x-1 =0 ,
x={2±sqrt(4+12)}/6.
x₁=1, x₂=-1/3 (※)
Ans: Coordinate of the point is X+x =1+1 =2
To determine the position where the third sphere, C, of charge +12 μC, should be placed so that there is no net force on it, we can use the principle of electrostatic equilibrium.
The net force on a charged object placed in an electric field is given by the equation:
F_net = F_1 + F_2 + ... + F_n
Where F_1, F_2, ..., F_n are the individual forces acting on the object due to each charged object present.
In this case, there are two charged objects, sphere A with a charge of +64 μC, and sphere B with a charge of -16 μC. The net force on sphere C will be zero when the individual forces due to sphere A and sphere B cancel each other out.
Now, let's calculate the individual forces acting on sphere C due to sphere A and sphere B.
1. Force due to sphere A on sphere C:
The force between two charged objects is given by Coulomb's law:
F = k*q1*q2/r^2
Where F is the force, k is the electrostatic constant (9 × 10^9 Nm^2/C^2), q1 and q2 are the charges of the respective objects, and r is the distance between the objects.
Since sphere A has a charge of +64 μC, the force due to sphere A on sphere C is:
F_A = k*q_A*q_C/r^2
where q_A = +64 μC and q_C = +12 μC
2. Force due to sphere B on sphere C:
The force due to sphere B on sphere C is given by the same equation as above:
F_B = k*q_B*q_C/r^2
where q_B = -16 μC and q_C = +12 μC
To find the position where the net force is zero, we need to set F_net = F_A + F_B = 0
Therefore:
F_A + F_B = 0
k*q_A*q_C/r^2 + k*q_B*q_C/r^2 = 0
Plugging in the known values, we have:
(9 × 10^9 Nm^2/C^2)(+64 μC)(+12 μC)/r^2 + (9 × 10^9 Nm^2/C^2)(-16 μC)(+12 μC)/r^2 = 0
Simplifying further, we can cancel out the common factors:
(+768 μC^2)(1/r^2) + (-192 μC^2)(1/r^2) = 0
576 μC^2(1/r^2) = 0
To solve for r^2, we can isolate it:
r^2 = (576 μC^2) / 0
As a denominator cannot be zero, we can conclude that there is no single position where the net force on sphere C is zero. The forces exerted by spheres A and B will always have some non-zero net force on sphere C.
To find the position where the third sphere, C, must be placed so that there is no net force on it, we need to consider the electric forces and use Coulomb's Law.
Coulomb's Law states that the electric force between two charged objects is proportional to the product of their charges and inversely proportional to the square of the distance between them.
First, let's calculate the electric force between spheres A and C. The force between two charges can be calculated using the formula:
F = k*q1*q2/r^2
Where F is the force between the charges, k is the electrostatic constant (approximately 9 x 10^9 N.m^2/C^2), q1 and q2 are the charges on the spheres, and r is the distance between them.
Using the given values:
Charge of sphere A, q1 = +64 μC
Charge of sphere C, q2 = +12 μC
Substituting these values into the formula, and assuming the distance between A and C is r1, we have:
F1 = k * q1 * q2 / r1^2
Next, let's calculate the electric force between spheres B and C. Assuming the distance between B and C is r2:
Charge of sphere B, q1 = -16 μC
Charge of sphere C, q2 = +12 μC
Using the same formula, we have:
F2 = k * q1 * q2 / r2^2
For there to be no net force on sphere C, the forces F1 and F2 must have equal magnitudes but opposite directions. Since the charges on spheres A, B, and C are already given, we need to find the distances r1 and r2 such that F1 = F2.
By equating the magnitudes of the two forces, we have:
k * q1 * q2 / r1^2 = k * q1 * q2 / r2^2
The charges q1 * q2, electrostatic constant k, and the distances r1 and r2 will cancel out, resulting in the following equation:
1 / r1^2 = 1 / r2^2
To solve for the position where sphere C must be placed, we can set r1^2 to be any value (e.g., 1) and then solve for r2^2:
1 / r1^2 = 1 / r2^2
1 / 1^2 = 1 / r2^2
1 = 1 / r2^2
Taking the reciprocal of both sides, we get:
r2^2 = 1
r2 = 1
Therefore, the distance between sphere B and C, r2, is 1 unit.
Thus, sphere C must be placed at a distance of 1 unit from sphere B on the positive x-axis to experience no net force.