Question
What is the electric potential at the center of a ring due to a point charge of +2.00 μC placed 2.00 m away and the ring itself with a charge of -3.00 μC and radius 0.500 m?
Answers
GPT-4o mini
To find the electric potential at the center of a ring due to a point charge and the ring itself, we analyze each contribution separately.
### Step 1: Contribution from the Point Charge
The electric potential \( V \) due to a point charge \( Q \) at a distance \( r \) is given by the formula:
\[
V = \frac{kQ}{r}
\]
where \( k \) is the Coulomb's constant, approximately \( 8.99 \times 10^9 \, \text{N m}^2/\text{C}^2 \).
For the point charge:
- Charge \( Q = 2.00 \, \mu\text{C} = 2.00 \times 10^{-6} \, \text{C} \)
- Distance \( r = 2.00 \, \text{m} \)
Substituting these values into the potential formula:
\[
V_{\text{point}} = \frac{(8.99 \times 10^9) \cdot (2.00 \times 10^{-6})}{2.00}
\]
Calculating this gives:
\[
V_{\text{point}} = \frac{(8.99 \times 10^9) \cdot (2.00 \times 10^{-6})}{2.00} = 8.99 \times 10^3 \, \text{V}
\]
### Step 2: Contribution from the Charged Ring
The electric potential \( V \) at the center of a uniformly charged ring is given by:
\[
V = \frac{kQ_{\text{ring}}}{R}
\]
For the ring:
- Charge \( Q_{\text{ring}} = -3.00 \, \mu\text{C} = -3.00 \times 10^{-6} \, \text{C} \)
- Radius \( R = 0.500 \, \text{m} \) (the distance from the ring to the center where we are calculating the potential)
Now, substituting these values:
\[
V_{\text{ring}} = \frac{(8.99 \times 10^9) \cdot (-3.00 \times 10^{-6})}{0.500}
\]
Calculating this gives:
\[
V_{\text{ring}} = \frac{(8.99 \times 10^9) \cdot (-3.00 \times 10^{-6})}{0.500} = -5.394 \times 10^4 \, \text{V}
\]
### Step 3: Total Electric Potential at the Center
Now we add the potentials from the point charge and the ring:
\[
V_{\text{total}} = V_{\text{point}} + V_{\text{ring}}
\]
Substituting the values:
\[
V_{\text{total}} = 8.99 \times 10^3 + (-5.394 \times 10^4)
\]
Calculating this gives:
\[
V_{\text{total}} = 8.99 \times 10^3 - 5.394 \times 10^4 = 8.99 \times 10^3 - 53940 = -44951 \, \text{V}
\]
Therefore, the total electric potential at the center of the ring due to both the point charge and the charged ring is approximately:
\[
\boxed{-44.951 \, \text{kV}}
\]
### Step 1: Contribution from the Point Charge
The electric potential \( V \) due to a point charge \( Q \) at a distance \( r \) is given by the formula:
\[
V = \frac{kQ}{r}
\]
where \( k \) is the Coulomb's constant, approximately \( 8.99 \times 10^9 \, \text{N m}^2/\text{C}^2 \).
For the point charge:
- Charge \( Q = 2.00 \, \mu\text{C} = 2.00 \times 10^{-6} \, \text{C} \)
- Distance \( r = 2.00 \, \text{m} \)
Substituting these values into the potential formula:
\[
V_{\text{point}} = \frac{(8.99 \times 10^9) \cdot (2.00 \times 10^{-6})}{2.00}
\]
Calculating this gives:
\[
V_{\text{point}} = \frac{(8.99 \times 10^9) \cdot (2.00 \times 10^{-6})}{2.00} = 8.99 \times 10^3 \, \text{V}
\]
### Step 2: Contribution from the Charged Ring
The electric potential \( V \) at the center of a uniformly charged ring is given by:
\[
V = \frac{kQ_{\text{ring}}}{R}
\]
For the ring:
- Charge \( Q_{\text{ring}} = -3.00 \, \mu\text{C} = -3.00 \times 10^{-6} \, \text{C} \)
- Radius \( R = 0.500 \, \text{m} \) (the distance from the ring to the center where we are calculating the potential)
Now, substituting these values:
\[
V_{\text{ring}} = \frac{(8.99 \times 10^9) \cdot (-3.00 \times 10^{-6})}{0.500}
\]
Calculating this gives:
\[
V_{\text{ring}} = \frac{(8.99 \times 10^9) \cdot (-3.00 \times 10^{-6})}{0.500} = -5.394 \times 10^4 \, \text{V}
\]
### Step 3: Total Electric Potential at the Center
Now we add the potentials from the point charge and the ring:
\[
V_{\text{total}} = V_{\text{point}} + V_{\text{ring}}
\]
Substituting the values:
\[
V_{\text{total}} = 8.99 \times 10^3 + (-5.394 \times 10^4)
\]
Calculating this gives:
\[
V_{\text{total}} = 8.99 \times 10^3 - 5.394 \times 10^4 = 8.99 \times 10^3 - 53940 = -44951 \, \text{V}
\]
Therefore, the total electric potential at the center of the ring due to both the point charge and the charged ring is approximately:
\[
\boxed{-44.951 \, \text{kV}}
\]