Question

A charge \(\mathrm{Q}\) is divided into two parts and then they are placed at a fixed distance. The force between the two charges is always maximum when the charges are \(\ldots \ldots\) (A) \((Q / 3),(Q / 3)\) (B) \((\mathrm{Q} / 2),(\mathrm{Q} / 2)\) (C) \((Q / 4),(3 Q / 4)\) (D) \((Q / 5),(4 Q / 5)\)

Short answer

The optimal way to divide the charge is to have both charges equal. Therefore, the charges should be \(\frac{Q}{2}\) and \(\frac{Q}{2}\), as in option (B).

Step-by-step solution

Understand the problem

We are given a charge Q which is divided into two parts say q1 and q2 such that q1+q2=Q. We're tasked to find in what ratio the charges should be divided so as to maximize the force between them.

Coulomb's Law

Coulomb's Law states that the force between two charges is directly proportional to the product of the charges and inversely proportional to the square of the distance between them. Mathematically, it can be written as: \[F = k\frac{q1 \cdot q2}{r^2}\] where \(F\) is the force between the charges, \(k\) is Coulomb's constant, \(q1\) and \(q2\) are the charges, and \(r\) is the distance between the charges.

Express q2 in terms of q1

Since q1 + q2 = Q, we can express q2 in terms of q1 as follows: \[q2 = Q - q1\]

Substitute q2 in the Coulomb's Law equation

We can now replace q2 in the Coulomb's Law equation with the expression we found above: \[F = k\frac{q1 \cdot (Q - q1)}{r^2}\]

Maximize the Force

For the force to be maximum, the product q1 * q2 should be maximum. We can rewrite the equation for \(F\) and eliminate the constant values: \[F' = q1 \cdot (Q - q1)\]

Find the first derivative of F'

To find the maximum value, we need to find the first derivative of F' with respect to q1 and set it to zero: \[\frac{dF'}{dq1} = (Q - q1) - q1 = Q - 2q1\]

Set the derivative to zero and solve for q1

Set the first derivative equal to zero and solve for q1: \[Q - 2q1 = 0\] \[q1 = \frac{Q}{2}\]

Substitute q1 back into q2 expression

Now, we can substitute the value of q1 back into the expression for q2: \[q2 = Q - \frac{Q}{2} = \frac{Q}{2}\] Therefore the answer is (B) \((\mathrm{Q} / 2),(\mathrm{Q} / 2)\).

Key concepts

Electric Charge

An electric charge is a fundamental property of matter that leads to the electromagnetic forces between particles. There are two types of electric charges: positive and negative. Likewise, like charges repel each other, while unlike charges attract each other. The unit for electric charge is the coulomb (C).

Electric charges are integral to understand how particles interact through the electromagnetic force. For example:

• An electron carries a negative charge of approximately \(-1.6 \times 10^{-19}\) coulombs.
• A proton carries a positive charge of the same magnitude.

Adjusting the quantity and distribution of charges in a system impacts the force experienced and exerted among the charges. Analyzing exercises involving charge division and force calculation helps to better understand such interactions.

Maximizing Force

When two electric charges are present, the force between them is determined by Coulomb's Law. This law helps us calculate and predict the interaction strength between charged particles:

• Force is maximized when the product of the charges is maximized.
• The charges should be placed at a fixed distance for an accurate measurement of maximum force.

To find this optimal force, you can use calculus and differentiate the equation of force with respect to one of the charge quantities to find a critical point. By setting the derivative to zero and solving, we can identify the condition for maximum force. In this case, equal charges \( \left( \frac{Q}{2}, \frac{Q}{2} \right) \) result in the maximum possible force.

Charge Distribution

Charge distribution involves dividing a total charge, \(Q\), into parts that can interact with each other over a fixed distance. The distribution can strongly influence the interaction force:

• Uniform distribution can lead to maximizing the force, which is the focus in many physics problems.
• Understanding how to distribute the charge can help solve problems more efficiently.

In this exercise, we distributed charge \(Q\) into two parts, \(q_1\) and \(q_2\), ensuring their sum equals the total charge \(Q\). By applying calculus, we deduce that an equal distribution \( \left( \frac{Q}{2}, \frac{Q}{2} \right) \) results in a maximized force between the charges, a key insight from Coulomb’s Law applications.