Question
Grade 12 Physics - Assignment #3 - Electricity
Use Grass Format for each question.
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.
Use Grass Format for each question.
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.
Answers
GPT-4o mini
### Assignment #3 - Electricity
**a)** To find the ratio \( \frac{F_{e1}}{F_{e2}} \), we will utilize Coulomb's Law, which states:
\[
F = k \frac{|q_1 q_2|}{r^2}
\]
where \( k \) is Coulomb's constant, \( q_1 \) and \( q_2 \) are the charges, and \( r \) is the distance between the charges.
1. **Initial Force \( F_{e1} \)**:
In this case, the charges are \( +q \) and \( -3q \). Therefore:
\[
F_{e1} = k \frac{|q \cdot (-3q)|}{r^2} = k \frac{3q^2}{r^2}
\]
Since they have opposite charges, \( F_{e1} \) is attractive.
2. **Final Force \( F_{e2} \)**:
When the spheres touch, they share their charges. The total charge becomes:
\[
+q + (-3q) = -2q
\]
Since the spheres are identical, the charge on each sphere after touching is:
\[
\text{Charge on each sphere} = \frac{-2q}{2} = -q
\]
Now, with both charges being \( -q \), the force is:
\[
F_{e2} = k \frac{|-q \cdot -q|}{r^2} = k \frac{q^2}{r^2}
\]
The ratio of the forces is:
\[
\frac{F_{e1}}{F_{e2}} = \frac{k \frac{3q^2}{r^2}}{k \frac{q^2}{r^2}} = \frac{3q^2}{q^2} = 3
\]
### **b)** Description of the Ratio
The ratio \( \frac{F_{e1}}{F_{e2}} = 3 \) indicates that the initial attractive force \( F_{e1} \) is three times stronger than the final repulsive force \( F_{e2} \). This means that while the spheres attract each other with a significant force, they do not repel each other with the same magnitude after touching.
### **c)** Attraction vs. Repulsion
Initially, the two spheres attract each other because they have unlike charges: one has a positive charge (+q) and the other has a negative charge (-3q). After they touch, they both acquire the same negative charge (-q), which leads to a repulsive force because like charges repel each other.
### **d)** New Charges on Each Sphere
As explained previously, the total charge after touching is:
\[
+q + (-3q) = -2q
\]
Dividing this charge equally between the two spheres results in:
\[
\text{Charge on each sphere} = \frac{-2q}{2} = -q
\]
So, each sphere ends up with a charge of \( -q \).
### **e)** Vector Diagrams
1. **Initial Attractive Force \( \mathbf{F_{e1}} \)**:
- Arrow pointing towards the center, indicating attraction between the two spheres (\( +q \) to \( -3q \)).
2. **Final Repulsive Force \( \mathbf{F_{e2}} \)**:
- Arrow pointing away from the center, indicating repulsion between the two identical charges (\( -q \) to \( -q \)).
### **f)** Change in Direction of Electric Force
The direction of electric force changes from attracting to repelling as the charges on the spheres change from unlike (attracting) to like (repelling) upon contact. Coulomb's law dictates that forces between unlike charges are attractive, while forces between like charges are repulsive. Thus, \( \mathbf{F_{e1}} \) acts towards each other, while \( \mathbf{F_{e2}} \) acts away from each other.
### **g)** Effect of Mass Disparity on Motion
If one of the spheres is more massive than the other, the interaction will still be governed by the same electrostatic forces, but the resulting motion will differ due to the difference in mass. According to Newton's second law (\( F = ma \)), the acceleration of each sphere will differ:
- The lighter sphere will accelerate more than the heavier sphere under the influence of the same electrostatic force, leading to different velocities. The motion is influenced by the mass ratio, with the lighter sphere moving away faster and the heavier sphere moving slower, but both moving in accordance with the direction of the electric forces.
**a)** To find the ratio \( \frac{F_{e1}}{F_{e2}} \), we will utilize Coulomb's Law, which states:
\[
F = k \frac{|q_1 q_2|}{r^2}
\]
where \( k \) is Coulomb's constant, \( q_1 \) and \( q_2 \) are the charges, and \( r \) is the distance between the charges.
1. **Initial Force \( F_{e1} \)**:
In this case, the charges are \( +q \) and \( -3q \). Therefore:
\[
F_{e1} = k \frac{|q \cdot (-3q)|}{r^2} = k \frac{3q^2}{r^2}
\]
Since they have opposite charges, \( F_{e1} \) is attractive.
2. **Final Force \( F_{e2} \)**:
When the spheres touch, they share their charges. The total charge becomes:
\[
+q + (-3q) = -2q
\]
Since the spheres are identical, the charge on each sphere after touching is:
\[
\text{Charge on each sphere} = \frac{-2q}{2} = -q
\]
Now, with both charges being \( -q \), the force is:
\[
F_{e2} = k \frac{|-q \cdot -q|}{r^2} = k \frac{q^2}{r^2}
\]
The ratio of the forces is:
\[
\frac{F_{e1}}{F_{e2}} = \frac{k \frac{3q^2}{r^2}}{k \frac{q^2}{r^2}} = \frac{3q^2}{q^2} = 3
\]
### **b)** Description of the Ratio
The ratio \( \frac{F_{e1}}{F_{e2}} = 3 \) indicates that the initial attractive force \( F_{e1} \) is three times stronger than the final repulsive force \( F_{e2} \). This means that while the spheres attract each other with a significant force, they do not repel each other with the same magnitude after touching.
### **c)** Attraction vs. Repulsion
Initially, the two spheres attract each other because they have unlike charges: one has a positive charge (+q) and the other has a negative charge (-3q). After they touch, they both acquire the same negative charge (-q), which leads to a repulsive force because like charges repel each other.
### **d)** New Charges on Each Sphere
As explained previously, the total charge after touching is:
\[
+q + (-3q) = -2q
\]
Dividing this charge equally between the two spheres results in:
\[
\text{Charge on each sphere} = \frac{-2q}{2} = -q
\]
So, each sphere ends up with a charge of \( -q \).
### **e)** Vector Diagrams
1. **Initial Attractive Force \( \mathbf{F_{e1}} \)**:
- Arrow pointing towards the center, indicating attraction between the two spheres (\( +q \) to \( -3q \)).
2. **Final Repulsive Force \( \mathbf{F_{e2}} \)**:
- Arrow pointing away from the center, indicating repulsion between the two identical charges (\( -q \) to \( -q \)).
### **f)** Change in Direction of Electric Force
The direction of electric force changes from attracting to repelling as the charges on the spheres change from unlike (attracting) to like (repelling) upon contact. Coulomb's law dictates that forces between unlike charges are attractive, while forces between like charges are repulsive. Thus, \( \mathbf{F_{e1}} \) acts towards each other, while \( \mathbf{F_{e2}} \) acts away from each other.
### **g)** Effect of Mass Disparity on Motion
If one of the spheres is more massive than the other, the interaction will still be governed by the same electrostatic forces, but the resulting motion will differ due to the difference in mass. According to Newton's second law (\( F = ma \)), the acceleration of each sphere will differ:
- The lighter sphere will accelerate more than the heavier sphere under the influence of the same electrostatic force, leading to different velocities. The motion is influenced by the mass ratio, with the lighter sphere moving away faster and the heavier sphere moving slower, but both moving in accordance with the direction of the electric forces.