27. How far from a point charge will the potential be 100 V? At what distance will it be 2.00 x 10^2?

The electric potential \( V \) due to a point charge \( q \) at a distance \( r \) is given by the formula:

\[
V = \frac{kq}{r}
\]

where:
- \( V \) is the electric potential,
- \( k \) is Coulomb's constant (\( k = 8.99 \times 10^9 \, \text{N} \cdot \text{m}^2 / \text{C}^2 \)),
- \( q \) is the charge,
- \( r \) is the distance from the point charge.

### Part 1: Distance for \( V = 100 \, \text{V} \)

Rearranging the formula to solve for \( r \):

\[
r = \frac{kq}{V}
\]

Substitute \( V = 100 \, \text{V} \):

\[
r_{100} = \frac{kq}{100}
\]

### Part 2: Distance for \( V = 2.00 \times 10^2 \, \text{V} \)

Substitute \( V = 200 \, \text{V} \):

\[
r_{200} = \frac{kq}{200}
\]

Thus, the ratio of the distances \( r_{100} \) and \( r_{200} \) will be:

\[
\frac{r_{100}}{r_{200}} = \frac{kq / 100}{kq / 200} = \frac{200}{100} = 2
\]

So, the distance for the potential being \( 100 \, \text{V} \) will be **twice** the distance for \( 200 \, \text{V}**.

Do you want to plug in a specific value for the charge \( q \), or keep this relation general? Let me know! 

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**Relative questions:**
1. What is the relationship between potential and distance for a point charge?
2. How does doubling the distance from a charge affect the potential?
3. What is Coulomb's constant, and why is it important in calculating electric potential?
4. How does the value of the charge \( q \) affect the distances where potential takes specific values?
5. How does electric potential differ from electric field in relation to distance?

**Tip:** Always keep track of units when working with electrostatics problems, especially with constants like Coulomb's constant.

How far from a point charge will the potential be 100 V? At what distance will it be 2.00 x 10^2?

Learn how to calculate the distance from a point charge where the electric potential is 100V and 200V. This problem covers key electrostatics concepts, including Coulomb's constant and the inverse relationship between potential and distance for a point charge. Suitable for Grades 11-12 students.

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