Question

A bead with charge \(q_{1}=1.27 \mu \mathrm{C}\) is fixed in place at the end of a wire that makes an angle of \(\theta=51.3^{\circ}\) with the horizontal. A second bead with mass \(m_{2}=3.77 \mathrm{~g}\) and a charge of \(6.79 \mu \mathrm{C}\) slides without friction on the wire. What is the distance \(d\) at which the force of the Earth's gravity on \(m_{2}\) is balanced by the electrostatic force between the two beads? Neglect the gravitational interaction between the two beads.

Short answer

Answer: The distance between the two charged beads when the gravitational force is balanced by the electrostatic force is 0.0206 m or 2.06 cm.

Step-by-step solution

Calculate the electrostatic force using Coulomb's law

Coulomb's law gives the force exerted between two charges as \( F_{e} = k \frac{q_1 q_2}{r^2} \), where: - \(F_{e}\) is the electrostatic force, - \(k = 8.9875 \times 10^9 Nm^2C^{-2}\) (Coulomb's constant), - \(q_1 = 1.27 \times 10^{-6} C\) (charge of the fixed bead), - \(q_2 = 6.79 \times 10^{-6} C\) (charge of the sliding bead), and - \(r\) is the distance between the charges.

Calculate the gravitational force on the second bead

The gravitational force on the second bead can be calculated using the equation: \( F_{g} = m_2 g \), where \(g=9.81 m/s^2\) is the gravitational acceleration and \(m_2=3.77\times10^{-3}kg\) is the mass of the second bead.

Set up an equilibrium equation

Since the electrostatic force between the beads should balance the gravitational force on the second bead when it slides on the wire, we can write the equilibrium equation as: \(\sin(\theta) F_{e} = F_{g}\). Plugging in Coulomb's law and the gravitational force equation, we get: \(\sin(\theta) \left( k \frac{q_1 q_2}{r^2} \right) = m_2 g \)

Solve for distance 'd'

Now we can solve for 'r' (since 'r' = 'd' in this case): \(\sin(\theta) \left( k \frac{q_1 q_2}{r^2} \right) = m_2 g \) Solving for 'r', we get: \( r = \sqrt{ k \frac{q_1 q_2}{m_2 g \sin(\theta)} } \) Plugging in the given values: \( r = \sqrt{ 8.9875 \times 10^9 \frac{1.27 \times 10^{-6} (6.79 \times 10^{-6})}{(3.77\times10^{-3})(9.81)\sin(51.3)} }\) Which after solving evaluates to: \(r = 0.0206 m\) Thus, the distance 'd' at which the electrostatic force balances the gravitational force is \(d=0.0206 m\) or \(2.06 cm\).

Key concepts

Electrostatic Force

Electrostatic force is a fundamental interaction between charged particles. According to Coulomb's Law, this force can be attractive or repulsive, depending on the nature of the charges involved. The force between two point charges, such as the beads in our exercise, is given by the formula:\[ F_e = k \frac{q_1 q_2}{r^2} \]Here, \(F_e\) represents the electrostatic force, \(k\) is the Coulomb's constant \( (8.9875 \times 10^9 \, \text{Nm}^2\text{C}^{-2}) \), and \(r\) is the distance between the charges, \(q_1\) and \(q_2\).

• The magnitude of the force is directly proportional to the product of the magnitudes of the charges.
• It is inversely proportional to the square of the distance between the charges.

This means that if you double the distance between the charges, the force decreases to one-fourth. Electrostatic force plays a crucial role in maintaining the structure of atoms and even in everyday phenomena like static electricity, where you might feel a small shock or see sparks.

Gravitational Force

Gravitational force is the attraction between objects with mass. It is the force that gives us weight and keeps planets in orbit. For the second bead in our problem, this force is calculated using the equation:\[ F_g = m_2 g \]In this equation, \(F_g\) denotes the gravitational force, \(m_2\) is the mass of the object, and \(g\) is the acceleration due to gravity \((9.81 \text{ m/s}^2)\). Unlike electrostatic force, gravitational force is always attractive.

• The magnitude of the gravitational force is directly proportional to the mass of the objects involved.
• This force does not change with electrical properties like charge but is solely dependent on mass and distance.

Gravitational force is much weaker compared to electrostatic forces, but it acts over vast distances, making it the dominant force at astronomical scales.

Equilibrium in Physics

Equilibrium is a state where all the forces acting on an object are balanced, and it does not experience any linear acceleration. In physics, understanding equilibrium provides insight into how forces interact and how systems respond.In our scenario, equilibrium occurs when the electrostatic force exerted by one bead balances exactly with the gravitational force acting on the sliding bead. This can be expressed as:\[ \sin(\theta) F_e = F_g \]This equation indicates that the component of electrostatic force along the direction of the wire's angle must be equal to the gravitational force.

• Achieving equilibrium often requires careful calculation of involved forces.
• It is important in ensuring that objects or systems remain stable under various forces.

The concept of equilibrium is fundamental in designing structures and systems where stability is paramount, from bridges to spacecraft.