Problem 73 - University Physics with Modern Physics

Question

Two positive point charges \(Q\) are held fixed on the \(x\) -axis at \(x=a\) and \(x=-a .\) A third positive point charge \(q,\) with mass \(m,\) is placed on the \(x\) -axis away from the origin at a coordinate \(x\) such that \(|x| \ll a\) . The charge \(q\) , which is free to move along the \(x\) -axis, is then released. (a) Find the frequency of oscillation of the charge \(q\) .(Hint: Review the definition of simple harmonic motion in Section 13.2 Use the binomial expansion \((1+z)^{n}=1+\) \(n z+n(n-1) z^{2} / 2+\cdots,\) valid for the case \(|z|<1 . .\) (b) Suppose instead that the charge \(q\) were placed on the \(y\) -axis at a coordinate \(y\) such that \(|y| \ll a,\) and then released. If this charge is free to move anywhere in the \(x y\) -plane, what will happen to it? Explain your answer.

Step-by-Step Solution

Answer

The charge q oscillates along x with frequency \(f = \frac{1}{2\pi}\sqrt{\frac{4kQq}{ma^3}}\). On the y-axis, it will also oscillate due to symmetric force components.

Step 1: Understanding the Physical Scenario

We have a symmetrical configuration with two fixed charges, each of magnitude \(Q\), located at positions \(x = a\) and \(x = -a\). A third charge \(q\) is along the \(x\)-axis, near the origin, at position \(x\). This charge is allowed to move along the \(x\)-axis. We are to determine the frequency of oscillation of this charge when it is influenced by the electric forces exerted by the fixed charges.

Step 2: Expression for Electric Force

The net force on the charge \(q\) due to the charges \(Q\) is given by superposition of forces from each charge. By Coulomb's law, the force due to the charge at \(x=a\) is \(F_1 = \frac{kQq}{(a-x)^2}\) and for \(x=-a\) is \(F_2 = \frac{-kQq}{(a+x)^2}\). The net force \(F\) acting on \(q\) is the sum: \[F = \frac{kQq}{(a-x)^2} - \frac{kQq}{(a+x)^2}\].

Step 3: Use Binomial Expansion

Using the binomial expansion for small displacements \(|x| \ll a\), we expand \((a\pm x)^{-2}\) which results in: \((a-x)^{-2} \approx a^{-2}(1 + 2\frac{x}{a})\) and \((a+x)^{-2} \approx a^{-2}(1 - 2\frac{x}{a})\). Substituting and simplifying gives the net force: \[ F = \frac{4kQqx}{a^3} \].

Step 4: Relate Force to Simple Harmonic Motion

The expression \( F = \frac{4kQqx}{a^3} \) shows that the force is proportional to the displacement \(x\) with a negative sign indicating a restoring force, a characteristic of simple harmonic motion (SHM). From SHM principles, \( F = m\omega^2x \), this gives us \( \omega^2 = \frac{4kQq}{ma^3} \).

Step 5: Calculate the Frequency of Oscillation

The angular frequency \(\omega\) is related to the regular frequency \(f\) by \(\omega = 2\pi f\). Solving for \(f\), we find: \[ f = \frac{1}{2\pi}\sqrt{\frac{4kQq}{ma^3}} \].

Step 6: Evaluate Motion Along y-Axis

If the charge \(q\) is placed on the \(y\)-axis, the electric forces due to the two charges at the \(x\)-axis have symmetric components in the \(x\)-direction which cancel out, leaving only a resultant force in the \(y\)-direction that pulls the charge back toward the axis, similar to the behavior found when on the \(x\)-axis. Hence, the charge will oscillate.

Key Concepts

Electric Forces

Electric forces play a crucial role in determining the motion of point charges in space. These forces arise due to interactions between charged particles. The strength and direction of the forces depend on the amount of charge and the distance between them.

• The forces between two point charges can be calculated using Coulomb's law, which states that the force magnitude is directly proportional to the product of their charges and inversely proportional to the square of their separation distance.
• The formula is given by: \( F = \frac{k \cdot |Q_1 \cdot Q_2|}{r^2} \), where \( k \) is Coulomb's constant, \( Q_1 \) and \( Q_2 \) are the charges, and \( r \) is the distance between charges.
• The direction of the force depends on the nature of the charges; like charges repel each other while opposite charges attract.

In the scenario described, the third charge \( q \) is subject to forces from two fixed charges \( Q \) on the x-axis. Due to their symmetrical placement and the nature of electric forces, the net force can show distinct characteristics of simple harmonic motion (SHM) when the charge is displaced slightly.

Oscillation Frequency

Oscillation frequency, often referred to simply as frequency, denotes how often an oscillating system completes a cycle in a unit of time. In simple harmonic motion (SHM), it determines how many oscillations occur per second.

• The frequency \( f \) is given by \( f = \frac{1}{2\pi} \sqrt{\frac{k}{m}} \), where \( k \) is the stiffness or force constant and \( m \) is the mass of the oscillating object.
• In the case of electric forces, the force constant becomes dependent on charge and distance, as shown by the equation \( k = \frac{4 k Q q}{a^3} \).
• Thus, the frequency further becomes \( f = \frac{1}{2 \pi} \sqrt{\frac{4 k Q q}{m a^3}} \), illustrating how electric interactions influence SHM characteristics.

Finding the oscillation frequency in systems involving electric forces helps us understand the rhythmic, repetitive motion of charges as they react to restoring forces. This understanding is critical in physics, especially in fields addressing wave and particle dynamics.

Point Charges

Point charges are idealized charges that are assumed to be located at a single point in space. This simplification allows us to focus on essential charge interactions without worrying about the charge's physical extent.

• When calculating the electric forces or fields from point charges, we often treat them as having no dimensions, which simplifies the mathematics used in physics and engineering.
• Point charges interact by producing electric fields, influencing other charges within these fields according to principles like superposition and interaction laws such as Coulomb's law.
• This superposition means that the total electric force experienced by a charge is the vector sum of the individual forces from each point charge present.

In the provided exercise, by treating all charges as point charges, we can simplify the problem to focus on the core dynamics of oscillating motion and electric forces within a symmetric charge configuration. This makes complex problems much more manageable and highlights key physical concepts effectively.