Problem 12 - Physics

Question

A charge of \(-3.00 \mu \mathrm{C}\) is fixed at the center of a compass. Two additional charges are fixed on the circle of the compass (radius \(=0.100 \mathrm{~m}\) ). The charges on the circle are \(-4.00 \mu \mathrm{C}\) at the position due north and \(+5.00 \mu \mathrm{C}\) at the position due east. What is the magnitude and direction of the net electrostatic force acting on the charge at the center? Specify the direction relative to due east.

Step-by-Step Solution

Answer

The net force is 17.1 N at 38.7° south of due west.

Step 1: Understand the Setup

We have three charges: a charge at the center \(-3.00 \mu \mathrm{C}\), a charge to the north \(-4.00 \mu \mathrm{C}\), and a charge to the east \(+5.00 \mu \mathrm{C}\). The radius of the circle is \(0.100\,\mathrm{m}\). We need to calculate the net electrostatic force on the center charge.

Step 2: Calculate Force Due to Northern Charge

Using Coulomb's law, the force between the charges at the center and due north is calculated. \[ F_{N} = k \frac{|q_{1}q_{2}|}{r^2} = \left(8.99 \times 10^{9} \right) \frac{|-3.00 \times 10^{-6} \cdot -4.00 \times 10^{-6}|}{(0.100)^2} \] \[ F_{N} = 10.8\,\mathrm{N} \] This force is directed southward because both charges are negative and repel each other.

Step 3: Calculate Force Due to Eastern Charge

Similarly, calculate the force between the center charge and the charge due east:\[ F_{E} = k \frac{|q_{1}q_{3}|}{r^2} = \left(8.99 \times 10^{9} \right) \frac{|-3.00 \times 10^{-6} \cdot 5.00 \times 10^{-6}|}{(0.100)^2} \] \[ F_{E} = 13.5\,\mathrm{N} \] Since one charge is negative and the other is positive, this force is directed leftward (to the west) along the east-west line.

Step 4: Determine Resultant Force Magnitude

The two perpendicular forces \( F_{N} \) (southward) and \( F_{E} \) (westward) form a right triangle. Use the Pythagorean theorem to find the resultant force magnitude:\[ F_{R} = \sqrt{F_{N}^2 + F_{E}^2} = \sqrt{(10.8)^2 + (13.5)^2} \] \[ F_{R} = 17.1\,\mathrm{N} \]

Step 5: Calculate Resultant Force Direction

Determine the angle \( \theta \) of the resultant force with respect to due east (westward):\[ \theta = \tan^{-1}\left(\frac{F_{N}}{F_{E}}\right) = \tan^{-1}\left(\frac{10.8}{13.5}\right) \] \[ \theta = 38.7^{\circ} \] The net force is therefore \(17.1\,\mathrm{N}\) at an angle of \(38.7^{\circ}\) south of due west.

Key Concepts

Coulomb's Law

Coulomb's Law is a fundamental principle used to calculate the force between two point charges. This law states that the magnitude of the electrostatic force between two charges is directly proportional to the product of their magnitudes and inversely proportional to the square of the distance between them.
For our compass problem, this is expressed mathematically as:

• \( F = k \frac{|q_1 q_2|}{r^2} \)
• Where \( F \) is the force, \( k \) is Coulomb's constant \( (8.99 \times 10^9 \, \mathrm{N\cdot m^2/C^2}) \), \( q_1 \) and \( q_2 \) are the charges, and \( r \) is the distance between them.

This law helps us understand how charges influence each other, forming the basis for our calculations in electrostatic interactions.

Charge Interaction

Charge interaction involves understanding how different charges exert forces on each other. In this exercise, we have both positive and negative charges.

• Like charges (both positive or both negative) repel each other.
• Opposite charges (one positive, one negative) attract each other.

For instance, the central charge is repelled by the northern charge, as both are negative. Conversely, the central charge is attracted to the eastern charge, since one is negative and the other is positive. Understanding these interactions is crucial to applying Coulomb's Law correctly.

Vector Addition

Vector addition is essential in understanding how forces combine. In this context, the forces acting on the center charge are vectors due to their directions and magnitudes.
For our problem:

• We have a force to the south from the northern charge and a force to the west from the eastern charge.
• These forces can be represented as vectors on a coordinate plane.

To find the net force, we use vector addition. This involves combining the forces using geometric methods like the Pythagorean theorem since the forces are perpendicular. This gives us the resultant force in both magnitude and direction.

Resultant Force

The resultant force is the combined effect of individual forces acting on a charge. In our scenario, the forces from the north and east combine to form a single resultant force.

• First, calculate the magnitude using the Pythagorean theorem: \[ F_{R} = \sqrt{F_{N}^2 + F_{E}^2} \]
• Then, find the direction using trigonometry: \[ \theta = \tan^{-1}\left(\frac{F_{N}}{F_{E}}\right) \]

These calculations yield the total net force acting on the central charge, combining both effects into one vector.

Electricity

Electricity involves the study of charges and their interactions, as explored in this exercise. It is a fundamental aspect of physics, influencing various phenomena from static electricity to complex circuits.

• Electricity is the foundation for understanding both electrostatics and electrodynamics.
• The concepts of charge, force, and interaction are key components.

In this exercise, understanding how static charges interact helps students grasp more advanced concepts in electricity, paving the way for further studies in electromagnetism and electronics.