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How To Calculate The Electric Force Between Two Charges?

When two charges are close to each other or separated with certain distance, they experience electric force between them. So, the Electric Force is an attractive or repulsive force between the two charges or charged objectsThe force can be either attractive or repulsive force in which like charges repel and unlike charges attract.

The electric force between two charged objects can be calculated by using the Coulomb's law.

"Coulomb's law states that, the electrostatic force of attraction or repulsion between two charges is directly proportional to the magnitude of charges and inversely proportional to the square of the distance between them."

According to the coulomb's law, the electrostatic force between two charges is given by,

F = k.Q1.Q2/r^2 Newton

is a electrostatic constant or coulomb's constant which is equal to 1/4𝜋𝞮o = 9x10^9 nm^2/c^2.
Q1Q2 are the charges
r is the distance between two charges
F is a electrostatic force

The force between two charges is inversely proportional to the square of the distance between two charges which is characterized by Inverse Square Law. So the Coulomb's law is also similar to the Newton's law of gravitational force between the two objects which is given by,

F = G.m1.m2/r^2 Newton


Example Problem-1

Now suppose let's say Q1 = 3x10^-6C, Q2 = 6x10^-6C is separated at a distance r=4m, calculate the force between two charges.

From the coulomb's law, the force between two charges is given byF=k.Q1.Q2/r^2 

Substituting given values in the above formula we get,

9x10^9 nm^2/c^2 * 3x10^-6C * 6x10^-6C/(4m)^2 

F = 10.125x10^-3 N

So the electrostatic force between two charges is 10.125x10^-3 N. The positive sign indicates repulsive force, since both charges are like charges.

Example Problem-2

Two charges Q1 = -4x10^-6C, Q2 = 5x10^-6C separated by a distance r=0.5m, find the electric force between two charges.

From the coulomb's law, the force between two charges is given by, 

F = k.Q1.Q2/r^2 

Substituting given values in the above formula we get,

F9x10^9 nm^2/c^2 * -4x10^-6C * 5x10^-6C/(0.5m)^2 

F = - 0.72 N

The electrostatic force between two charges is -0.72 N. The negative sign indicates attractive force, since both charges are unlike charges.

Example Problem-3

Two charges Q1 = -3.5x10^-6C and Q2 = 5x10^-6C having a force of -0.045N. Calculate the distance between two charges.

From the coulomb's law we have, F=k.Q1.Q2/r^2 

Now, the distance between two charges is calculated by,

r^2 = k.Q1.Q2/F

= Sqrt(k.Q1.Q2/F)

Substituting given values in the above equation we get,

= Sqrt(9x10^9 nm^2/c^2 * -3.5x10^-6C * 5x10^-6C/-0.045N)

r = Sqrt(3.5) = 1.8708m

Hence the distance between two charges is 1.8708m

Example Problem-4

Two charges Q1 = -5x10^-6C, Q2 = 7x10^-6C separated by a distance r=0.6m. Now charge on the both objects is doubled, then what is the new electric force between the two charges?

From the coulomb's law, the force between two charges is given by, 

F = k.Q1.Q2/r^2 

Now since, charge on both the object is doubled so new force will be,

F = k.2(Q1).2(Q2)/r^2 

Substituting given values in the above formula we get,

F9x10^9 nm^2/c^2 * 2(-5x10^-6C ) * 2(7x10^-6C)/(0.6m)^2 

F = - 3.5 N

If charge on both object was not doubled, then force would be four times smaller which is - 0.875N 

Similarly, if the charge on any one of object is tripled, then new force will be,

F = k.3(Q1).Q2/r^2 

Substituting given values in the above formula we get,

9x10^9 nm^2/c^2 * 3(-5x10^-6C ) * (7x10^-6C)/(0.6m)^2 

F = - 2.625 N

Example Problem-5

Two charges Q1 = 2x10^-6C, Q2 = 4x10^-6C is separated at a distance of r=3m. Now the distance between both charges is doubled and what will be the new force between two charges?

From the coulomb's law, the force between two charges is given byF=k.Q1.Q2/r^2 

Since the distance between both charges is doubled then new force will be,

F=k.Q1.Q2/2r^2 

Substituting given values in the above formula we get,

9x10^9 nm^2/c^2 * 2x10^-6C * 4x10^-6C/2(3m)^2 

F = 4x10^-3 N


Quiz:

1. Two charges Q1 = -7x10^-6C, Q2= -2.5x10^-6C, the force between two charges will be attractive or repulsive?

2. Two charges Q1 = -2.5x10^-3C, Q2= 1C, the force between two charges will be attractive or repulsive?

3. Two charges with a repulsive force of 0.04N. If any one of the charge is doubled, then what will be the new force?

4. Two charges with a attractive force of -0.04N. And if the charge on both of the object is doubled, then what will be the new force?

5. Two charges with a repulsive force of 0.065N. If the distance between both charges is doubled, then what will be the new force?

6. Two charges with a attractive force of - 0.065N. If the distance between both charges is tripled, then what will be the new force?

7. Two charges with a repulsive force of 0.065N. If the distance between both charges is halved, then what will be the new force?

8. Two charges with repulsive force of 0.75N. If any one of the charge is doubled and distance separating the charges is also doubled, then what will be the new force?

9. Two charges with attractive force of - 0.75N. If both of the charges is doubled and distance separating the charges is also doubled, then what will be the new force?

10. Two charges with repulsive force of 0.75N. If both of the charges is tripled and distance separating the charges is also tripled, then what will be the new force?

11. Two charges with repulsive force of 0.05N. If the distance between the charges is quadrupled, then what will be the new force? 

12. Two charges with repulsive force of 0.03N. If the charges on both object is tripled and distance separated by them is halved, then what will be the new force?




Answer:

1. Repulsive force. Since both are like charges with same sign.

2. Attractive force. Since both are unlike charges with opposite sign.

3. 0.08N Since electrostatic force is directly proportional to the charge on each objects. If the charge on one object is doubled, then force also becomes double. So two times 0.04N is 0.08N.

4. -0.16N Since electrostatic force is directly proportional to the charge on each objects. If the charge on both object is doubled, then force becomes four times greater. So four times 0.04N is 0.16N.

5. 0.01625N Since electrostatic force is inversely proportional to the square of the distance. When the distance is doubled, then F becomes four times smaller. So one-fourth of 0.065 is 0.01625.

6. - 0.00723N Since electrostatic force is inversely proportional to the square of the distance. When the distance is tripled, then F becomes nine times smaller. So one-ninth of 0.065 is 0.00723.

7. 0.26N Since electrostatic force is inversely proportional to the square of the distance. When distance is halved, the F becomes four times larger. So four times 0.065N is 0.26N

8. 0.375N Since electrostatic force is directly proportional to the product of charges and inversely proportional to the square of the distance between them. So doubling one of the charge would double the force and doubling the distance would decrease the force by factor of four. Combining all these effect will reduce the force by a factor of two. So 0.75N/2 is 0.375N.

9. - 0.75N Here the charges on both object and the distance is doubled. So it will give the same result.

10. 0.75N Here the charges on both object and the distance is tripled. So it will also give the same result.

11. 0.003125N Since electrostatic force is inversely proportional to the square of the distance. If we increase the distance by four(quadrupled), the F becomes 16 times smaller. So one-sixteenth of 0.05N is 0.003125N.

12. 1.08N Tripling one of the charge will triple the force, but tripling both charges would increase force by factor of nine. (because two forces are multiplied here) and when the distance is halved, the F becomes four times larger. Combining all these effect would increase F by a factor of 36. So 36 times 0.03N is 1.08N.

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