Question

SPH4U Assignment 3 – Electricity
Every question should be done in Grass format.
Problem: Two small, identical spheres, with an initial charge of +q and -3q, respectively, attract each other with a force of 𝐹𝑒1→ when held a distance r apart. The two spheres are allowed to touch and are then drawn apart to the distance r. Now they repel with a force of 𝐹𝑒2 →.
a) Find the ratio 𝐹𝑒1 / 𝐹𝑒2 of the two forces. Show your work
b) Describe what this ratio means in terms of magnitude and direction of the two forces, 𝐹𝑒1→ and 𝐹𝑒2→.
c) Why do the two spheres attract initially, but repel after they are brought into contact?
d) After the spheres are touched and then separated, what is the new charge on each sphere? Show your work
e) Draw vector diagrams of the force vectors for both the initial attractive force and the final repulsive force. Clearly indicate the direction and relative magnitude of each.
f) Describe how the direction of the electric force changes from the initial interaction to after the spheres touch and are separated. Explain why this change in direction occurs, using Coulomb’s law and the concept of electric force vectors between like and unlike charges.
g) If one of the spheres were more massive than the other, how would this affect the motion of the spheres during their interaction? Assume the electrostatic forces are the same.

Answer 1

Certainly! Here’s how to structure your assignment in the Grass format, addressing each part systematically.

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### SPH4U Assignment 3 – Electricity

**Problem Statement:**
Two small, identical spheres have an initial charge of +q and -3q, respectively. They attract each other with a force \( \vec{F}_{e1} \) when held a distance \( r \) apart. After touching and being separated to the same distance \( r \), they repel each other with a force \( \vec{F}_{e2} \).

### a) Finding the Ratio \( \frac{F_{e1}}{F_{e2}} \)

**Initially:**
Using Coulomb’s Law:
\[
F_{e1} = k \frac{|q_1 \cdot q_2|}{r^2} = k \frac{|(+q)(-3q)|}{r^2} = k \frac{3q^2}{r^2}
\]

**After Touching:**
When the spheres touch, the total charge is \( +q - 3q = -2q \). They share this charge, resulting in:
\[
Q_{new} = \frac{-2q}{2} = -q
\]

Now, the force after separation is:
\[
F_{e2} = k \frac{|Q_{new1} \cdot Q_{new2}|}{r^2} = k \frac{|-q \cdot -q|}{r^2} = k \frac{q^2}{r^2}
\]

**Calculating the ratio:**
\[
\frac{F_{e1}}{F_{e2}} = \frac{k \frac{3q^2}{r^2}}{k \frac{q^2}{r^2}} = \frac{3q^2}{q^2} = 3
\]

### b) Meaning of the Ratio \( \frac{F_{e1}}{F_{e2}} \)

The ratio \( \frac{F_{e1}}{F_{e2}} = 3 \) indicates that the initial attractive force \( \vec{F}_{e1} \) is three times stronger than the final repulsive force \( \vec{F}_{e2} \). The direction of \( \vec{F}_{e1} \) is toward each other (attraction), while the direction of \( \vec{F}_{e2} \) is away from each other (repulsion).

### c) Reason for Force Direction Change

Initially, the charges are opposite (+q and -3q), leading to attraction. After being brought into contact, the charges equalize, resulting in like charges (-q each). Like charges repel each other, hence the force changes from attraction to repulsion.

### d) New Charge on Each Sphere

After touching, each sphere has a new charge:
\[
Q_{new} = \frac{-2q}{2} = -q
\]
Thus, after separation, each sphere has a charge of -q.

### e) Vector Diagrams

1. **For the Initial Attractive Force \( \vec{F}_{e1} \)**:

\[
[\text{Sphere 1 (+q)}] \quad \quad \vec{F}_{e1} \rightarrow \quad \quad [\text{Sphere 2 (-3q)}]
\]

(Direction: towards each other)

2. **For the Final Repulsive Force \( \vec{F}_{e2} \)**:

\[
[\text{Sphere 1 (-q)}] \quad \quad \vec{F}_{e2} \rightarrow \quad \quad [\text{Sphere 2 (-q)}]
\]

(Direction: away from each other)

### f) Change in Force Direction Explanation

The direction of the electric force changes due to the difference in charge types. Initially, opposite charges lead to attraction (as per Coulomb’s law, opposite charges attract). After touching, both have the same negative charge, resulting in repulsion (like charges repel).

### g) Effect of Mass Difference on Motion

If one sphere is more massive than the other, the motion during interaction will be influenced by the inertia of the more massive sphere. Assuming the electrostatic forces remain the same, the less massive sphere will experience a greater acceleration due to the same force, resulting in more noticeable motion towards or away from the other sphere.

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This structured approach allows for clear presentation of your work and reasoning, adhering to the requirements of your assignment.