Four-point charges Q, q, Q and q are placed at the corners of a square of side ‘a’ as shown in the figure.

Find the

(a) resultant electric force on a charge Q, and

(b) potential energy of this system.

OR

(a) Three-point charges q, – 4q and 2q are placed at the vertices of an equilateral triangle ABC of side ‘l’ as shown in the figure. Obtain the expression for the magnitude of the resultant electric force acting on the charge q.

(b) Find out the amount of the work done to separate the charges at infinite distance.

a) As seen from the diagram there will be three forces on the charge Q at one of the corners of the square.

Now we know that from coulomb’s law, force between two charges is given as

Where q₁ and q₂ are the charges separated by a distance r.

Applying coulomb’s law for charge Q(at C) and q (at B)

Applying coulomb’s law for charge Q(at C) and q (at D)

Applying coulomb’s law for charge Q(at C) and Q (at A)

Now from the diagram it can be seen that forces F₁ and F₂ are perpendicular to each other. So, the magnitude of their resultant will be

F’ = √F²₁ + F²₂ + F₁F₂cos90°

Since cos90° is zero and F₁ = F₂, therefore

F’ = √F²₁ + F²₂ = F₁√2

Now applying superposition principle, we get

F = F’ + F₃

b) Potential energy is between two charges is given as

Where q₁ and q₂ are the charges separated by a distance r.

Due to the symmetry of the system potential energy between charges that are adjacent to each other will be same.

Potential energy between diagonally opposite charge q

Potential energy between diagonally opposite charge Q

Therefore, the total potential energy of the system will be

U = U₁ + U₂ + U₃

OR

a) We know that from coulomb’s law, force between two charges is given as

Where q₁ and q₂ are the charges separated by a distance r.

Applying coulomb’s law between charge -4q and q

Applying coulomb’s law for charge 2q and q

From the diagram we can see that the angle between F₁ and F₂ is 120°. So, magnitude of their resultant force F will be

F = √F²₁ + F²₂ + F₁F₂cos90

Since F₁ = 2F₂ and cos120° = -1/2

F = √(2F₂)² + F²₂ + 4F²₂cos120°

F = √4 F²₂ + F²₂-2 F²₂

F = √3 F²₂

F = F₂√3

So, the net electric force on charge q is

b) The amount of work done in separating the charges to infinity will be equal to potential energy.

Potential energy is between two charges is given as

Where q1 and q2 are the charges separated by a distance r.

The total potential energy of the system