Question

Four equal charges \(\mathrm{Q}\) are placed at the four corners of a square of each side is ' \(\mathrm{a}\) '. Work done in removing a charge \- Q from its centre to infinity is ....... (A) 0 (B) \(\left[\left(\sqrt{2} \mathrm{Q}^{2}\right) /\left(\pi \epsilon_{0} \mathrm{a}\right)\right]\) (C) \(\left[\left(\sqrt{2} Q^{2}\right) /\left(4 \pi \epsilon_{0} a\right)\right]\) (D) \(\left[\mathrm{Q}^{2} /\left(2 \pi \epsilon_{0} \mathrm{a}\right)\right]\)

Short answer

The work done in removing a charge -Q from the center of the square to infinity is \(W = \left[\left(\sqrt{2} Q^{2}\right) /\left(4 \pi \epsilon_{0} a\right)\right]\).

Step-by-step solution

Calculate the electric potential at the center of the square

To calculate the electric potential at the center of the square, first, we need to find the distance between the center and one of the corners. Recalling that the diagonal of a square can be found using the Pythagorean theorem, we have: \(d = \sqrt{a^2 + a^2}\) As all sides of the square are equal in length, we have: \(d = \sqrt{2a^2} = a\sqrt{2}\) Now, we can find the electric potential at the center due to one of the corner charges using the formula: \(V = \frac{kQ}{r}\) Where \(k = \frac{1}{4πε_0}\) and \(r = a\sqrt{2}\). We have four equal charges Q at the corners, so the total potential at the center (V_total) is: \(V_{total} = 4 \times \frac{kQ}{a\sqrt{2}}\)

Calculate the work done to move the negative charge to infinity

The work done to move a charge against an electric field is given by the product of the charge and the electric potential difference. In this case, the potential difference is given by (V_total - 0) since the potential at infinity is 0. Therefore, the work done (W) is given by: \(W = (-Q) \times \left(4 \times \frac{kQ}{a\sqrt{2}} - 0\right)\) Simplify the expression: \(W = \left(-\frac{4kQ^2}{a\sqrt{2}}\right)\) Now, substituting the value of \(k = \frac{1}{4πε_0}\): \(W = \left(\frac{-4Q^2}{4πε_0a\sqrt{2}}\right)\) Thus, the work done in removing a charge -Q from the center of the square to infinity is: \(W = \left[\left(\sqrt{2} Q^{2}\right) /\left(4 \pi \epsilon_{0} a\right)\right]\) So, the correct answer is (C).

Key concepts

Work Done

Work done is a concept in physics that quantifies the energy transferred when an object moves under the influence of a force. In electrostatics, when a charge is moved in an electric field, work must be done against the field’s force.
In our problem, we calculate the work done in moving a charge (-Q) from the center of a square of side 'a' to infinity.
This involves understanding the electric potential at the start point and noting that at infinity, the potential is zero. The formula for work done in this scenario is given by \( W = q \Delta V \), where \( \Delta V \) is the potential difference.

• This illustrates the energy required to overcome the electric forces acting on the charge.
• The work done is negative, indicating energy is released by the system when the charge moves.

Charge Configuration

Charge configuration refers to the arrangement of charges in a given system. In this exercise, we have four equal positive charges \(Q\) placed at the corners of a square with side length 'a'.
The symmetrical arrangement of these charges means each charge exerts equal force at the center, resulting in an even distribution of potential.

• Each charge contributes to the electric potential at a point, allowing us to calculate the total potential.
• The symmetry simplifies the calculation, as potential from each charge can be summed up linearly.
• Understanding the configuration helps to simplify complex problems by recognizing symmetrical properties.

Electric Field

An electric field is a region around a charged particle where a force would be exerted on other charges. It's an crucial concept in understanding how forces act at a distance in electrostatic problems.
In the problem, understanding the electric field is key to calculating the potential and hence the work done on moving the -Q charge.

• Electric potential at a point is related to the work done against an electric field to bring a unit charge from infinity to that point.
• For a point charge, the electric field decreases with distance, shown by \( E = \frac{kQ}{r^2} \).
• Knowing the field helps in understanding how energy is stored and transferred in electric systems.

Pythagorean Theorem

The Pythagorean theorem is a fundamental relation in geometry, used often in physics problems to calculate distances in a plane. It states that for a right-angled triangle, the sum of the squares of the two shorter sides is equal to the square of the hypotenuse.
In our exercise, it helps to determine the distance from the center of the square to any of its corners.

• The diagonal of the square is calculated using \( d = \sqrt{a^2 + a^2} = a\sqrt{2} \).
• This distance is critical for finding the electric potential due to a charge at the square’s corners.
• Using geometry effectively allows us to translate spatial configurations into calculable problems.