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Chapter 17: Problem 30

\(\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?

Short Answer

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

Step by step solution

01

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.
02

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.
03

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.
04

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:\[ \frac{|q|}{x^2} = \frac{4Q}{d^2} \]
05

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 \(+4Q\).
06

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 \(q = -16Q\).
07

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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

\(\bullet\) A negative charge of \(-0.550 \mu C\) exerts an upward 0.200 \(\mathrm{N}\) force on an unknown charge 0.300 \(\mathrm{m}\) directly below it. (a) What is the unknown charge (magnitude and sign)? (b) What are the magnitude and direction of the force that the unknown charge exerts on the - 0.550\(\mu \mathrm{C}\) charge?

\(\bullet$$\bullet \mathrm{A}-5.00 \mathrm{nC}\) point charge is on the \(x\) axis at \(x=1.20 \mathrm{m} . \mathrm{A}\) second point charge \(Q\) is on the \(x\) axis at \(-0.600 \mathrm{m} .\) What must be the sign and magnitude of \(Q\) for the resultant electric field at the origin to be (a) 45.0 \(\mathrm{N} / \mathrm{C}\) in the \(+x\) direction, \((\mathrm{b}) 45.0 \mathrm{N} / \mathrm{C}\) in the \(-x\) direction?

\(\bullet$$\bullet\) In a certain region of space, the electric field \(E\) is uniform; i.e., neither its direction nor its magnitude changes in the region. (a) Use Gauss's law to prove that this region of space must be electrically neutral; that is, there must be no charge in this region. (b) Is the converse true? That is, in a region of space where there is no charge, must \(\vec{E}\) be uniform? Explain.

\(\bullet\) As you walk across a synthetic-fiber rug on a cold, dry win- ter day, you pick up an excess charge of \(-55 \mu \mathrm{C}\) . (a) How many excess electrons did you pick up? (b) What is the charge on the rug as a result of your walking across it?

\(\bullet\) \(\bullet\) (a) An electron is moving east in a uniform electric field of 1.50 \(\mathrm{N} / \mathrm{C}\) directed to the west. At point \(A\) , the velocity of the electron is \(4.50 \times 10^{5} \mathrm{m} / \mathrm{s}\) toward the east. What is the speed of the electron when it reaches point \(B, 0.375 \mathrm{m}\) east of point \(A\) ? (b) A proton is moving in the uniform electric field of part (a). At point \(A\) , the velocity of the proton is \(1.90 \times 10^{4} \mathrm{m} / \mathrm{s},\) east. What is the speed of the proton at point \(B\) ?
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