Problem 30 - College Physics

Question

\(\bullet$$\bullet$$\bullet\) A charge \(+Q\) is located at the origin and a second charge, \(+4 Q,\) is at distance \(d\) on the \(x\) -axis. Where should a third charge, \(q,\) be placed, and what should be its sign and magnitude, so that all three charges will be in equilibrium?

Step-by-Step Solution

Answer

Place charge \(-16Q\) at \(x = -\frac{d}{2}\) to achieve equilibrium.

Step 1: Determine the Forces on Each Charge

For equilibrium, the net force on each charge should be zero. Consider only the forces due to other charges since the environment is unspecified. The force between two point charges is given by Coulomb's Law, \( F = \frac{{k \cdot |q_1 \cdot q_2|}}{{r^2}} \), where \( k \) is Coulomb's constant, \( q_1 \) and \( q_2 \) are the magnitudes of charges, and \( r \) is the distance between them.

Step 2: Analyze Forces on Charge +Q at the Origin

The forces acting on charge \(+Q\) are due to charges \(+4Q\) and \(q\). * Force due to \(+4Q\) is repulsive and acts along the positive x-axis.* Force due to \(q\) (yet to be defined) must be in the opposite direction to achieve equilibrium.

Step 3: Analyze Forces on Charge +4Q at Position x = d

The forces on charge \(+4Q\) are from \(+Q\) and \(q\). * Force due to \(+Q\) is repulsive and directed toward negative x-axis.* Force due to \(q\) should balance the force from \(+Q\) so it must act towards positive x-axis.

Step 4: Calculate Equilibrium Condition for Charge +Q

Assume charge \(q\) is at position \(x\).Equilibrium condition states: \( F_q = F_{4Q} \)Using Coulomb's law, equate the forces:\[ \frac{k \cdot Q \cdot |q|}{x^2} = \frac{k \cdot Q \cdot 4Q}{d^2} \]The \(k\) and \(Q\) cancel out:\[ \frac{|q|}{x^2} = \frac{4Q}{d^2} \]

Step 5: Solve for Position and Sign of Charge q

Assuming \(x = -a\) (to be left of \(+Q\) on x-axis), substitute \(x = -a\):\[ \frac{|q|}{a^2} = \frac{4Q}{d^2} \]Calculate \(a\) from balancing forces on all charges:\( a = \frac{d}{2} \).Since the charge \(q\) provides the opposite force direction on \(+Q\) and is left of it:\(q\) must be negative to balance the positive charges \(+Q\) and \(+4Q\).

Step 6: Determine the Magnitude of Charge q

Substitute \( a = \frac{d}{2}\) into the equilibrium condition:\[ \frac{|q|}{(\frac{d}{2})^2} = \frac{4Q}{d^2} \]Solve for \(|q|\):\[ |q| = \frac{4Q}{(\frac{d}{2})^2} \, \Rightarrow \, |q| = 16Q \]Thus, the required \(q = -16Q\).

Step 7: Verify Conditions for All Charges

Check if the derived position and charge magnitude ensure equilibrium for all charges by substituting back and confirming the net forces equate to zero.

Key Concepts

Electric Charge

Electric charge is a fundamental property of matter. It comes in two types: positive and negative. Charges interact with one another through forces described by Coulomb's Law. Coulomb's Law tells us the magnitude of the force between two point charges. It's determined by the formula:

• \( F = \frac{{k \, |q_1 \cdot q_2|}}{{r^2}} \)

Here, \( k \) is Coulomb's constant, \( q_1 \) and \( q_2 \) are the magnitudes of charges, and \( r \) is the distance between them. The force is stronger the closer the charges are and weaker the further apart they are. Like charges repel each other, while opposite charges attract.
This concept is crucial in many physics problems and applications, including this exercise where charges need to balance to achieve an equilibrium state.

Force Equilibrium

Force equilibrium occurs when all the forces acting on a system result in a net force of zero. For a system of charged particles, this means arranging the charges so that each one is subject to no unbalanced forces.
In this problem, we want the charges placed such that the net force on each charge is zero.

• The force on charge \(+Q\) due to other charges must be cancelled out by the force of another charge.
• Similarly, charge \(+4Q\) experiences two forces that should also be balanced.
• The third charge, \( q \), needs to be positioned so that it counteracts the repulsive forces experienced by each charge.

Achieving equilibrium means that these charges will remain at rest, assuming no other external forces act on the system.

Point Charges

Point charges are simplified models of charges where the size of each charge is considered to be insignificantly small. This allows us to focus purely on the mathematical description of electric forces without having to consider charge shape or distribution.
In equilibrium problems involving point charges like this one, you calculate forces based on the assumption that the charges are concentrated at a single point in space.

• It simplifies mathematical calculations, making it easier to apply Coulomb's Law.
• Assuming charges as point charges, we can ignore additional complexities like surface area effects.

In our exercise, all charges \(+Q\), \(+4Q\), and \(q\) are treated as point charges to simplify the calculations necessary for solving the problem.