Solve the equation for $$ q_{1} $$. $$ =\frac{k q_{1} q_{2}}{r^{2}} $$

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UpStudy AI · Accepted Answer

Certainly! It looks like you're working with Coulomb's Law, which describes the electrostatic force between two point charges. The standard form of Coulomb's Law is:

$$
F = \frac{k \, q_{1} q_{2}}{r^{2}}
$$

Where:
- $$ F $$ is the electrostatic force between the charges,
- $$ k $$ is Coulomb's constant ($$ k \approx 8.9875 \times 10^{9} \, \text{N·m²/C²} $$),
- $$ q_{1} $$ and $$ q_{2} $$ are the magnitudes of the two point charges,
- $$ r $$ is the distance between the centers of the two charges.

**Objective:** Solve for $$ q_{1} $$.

### Step-by-Step Solution

1. **Start with Coulomb's Law:**
   
   $$
   F = \frac{k \, q_{1} q_{2}}{r^{2}}
   $$

2. **Isolate $$ q_{1} $$ on one side:**

To solve for $$ q_{1} $$, you want to get $$ q_{1} $$ by itself. Follow these algebraic steps:

3. **Multiply both sides by $$ r^{2} $$:**
   
   $$
   F \cdot r^{2} = k \, q_{1} q_{2}
   $$

4. **Divide both sides by $$ k \, q_{2} $$ to solve for $$ q_{1} $$:**
   
   $$
   q_{1} = \frac{F \cdot r^{2}}{k \, q_{2}}
   $$

### Final Expression

$$
q_{1} = \frac{F \, r^{2}}{k \, q_{2}}
$$

### Example Calculation

Suppose you have the following values:
- Electrostatic force, $$ F = 1.0 \times 10^{-5} \, \text{N} $$
- Distance between charges, $$ r = 0.05 \, \text{m} $$
- Coulomb's constant, $$ k = 8.9875 \times 10^{9} \, \text{N·m²/C²} $$
- Charge $$ q_{2} = 2.0 \times 10^{-6} \, \text{C} $$

Plugging these into the equation:

$$
q_{1} = \frac{(1.0 \times 10^{-5} \, \text{N}) \times (0.05 \, \text{m})^{2}}{(8.9875 \times 10^{9} \, \text{N·m²/C²}) \times (2.0 \times 10^{-6} \, \text{C})}
$$

$$
q_{1} = \frac{1.0 \times 10^{-5} \times 0.0025}{8.9875 \times 10^{9} \times 2.0 \times 10^{-6}}
$$

$$
q_{1} = \frac{2.5 \times 10^{-8}}{1.7975 \times 10^{4}}
$$

$$
q_{1} \approx 1.39 \times 10^{-12} \, \text{C}
$$

**So, $$ q_{1} \approx 1.39 \times 10^{-12} \, \text{C} $$.**

### Summary

To solve for $$ q_{1} $$ in Coulomb's Law, rearrange the equation as follows:

$$
q_{1} = \frac{F \, r^{2}}{k \, q_{2}}
$$

Ensure that all units are consistent when plugging in numerical values to obtain the correct result.
To solve for $$ q_{1} $$ in Coulomb's Law, use the equation: $$ q_{1} = \frac{F \, r^{2}}{k \, q_{2}} $$ Where: - $$ F $$ is the electrostatic force, - $$ r $$ is the distance between charges, - $$ k $$ is Coulomb's constant, - $$ q_{2} $$ is the other charge. Plugging in the values will give you the value of $$ q_{1} $$.

Solve the equation for \( q_{1} \). \[ =\frac{k q_{1} q_{2}}{r^{2}} \]

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