8. Charge $$ Q_{A}=-20 \mu C $$ is located at $$ A(-6,4,7) $$ and change $$ Q_{B}=50 \mu C $$ is at $$ B(5,8-2) $$ free space The vector force excented on $$ Q_{A} $$ by $$ Q_{B} $$ is $$ \text { Columb laws }: F=\frac{q_{1} q_{2}}{4 \pi \varepsilon_{0} R^{2}} \cdot \overrightarrow{q^{R}} $$ Where $$ K $$ is coulomb constant $$ \left(8.99 \times 10^{9} \mathrm{~N} \mathrm{M}\right. $$

Question

UpStudy AI · Accepted Answer

To determine the vector force exerted on charge $$ Q_A $$ by charge $$ Q_B $$ using Coulomb's Law, we'll follow these steps:

### Given:
- **Charge $$ Q_A $$**: $$ -20 \, \mu C $$ located at point $$ A(-6, 4, 7) $$
- **Charge $$ Q_B $$**: $$ 50 \, \mu C $$ located at point $$ B(5, 8, -2) $$
- **Coulomb's Constant $$ k $$**: $$ 8.99 \times 10^{9} \, \text{N} \cdot \text{m}^2/\text{C}^2 $$

### Coulomb's Law:
$$
\vec{F} = k \frac{Q_A Q_B}{R^3} \vec{R}
$$
where:
- $$ \vec{R} $$ is the displacement vector from $$ Q_A $$ to $$ Q_B $$
- $$ R $$ is the magnitude of $$ \vec{R} $$

### Step 1: Calculate the Displacement Vector $$ \vec{R} $$
$$
\vec{R} = B - A = (5 - (-6), \, 8 - 4, \, -2 - 7) = (11, \, 4, \, -9)
$$

### Step 2: Calculate the Distance $$ R $$
$$
R = \sqrt{11^2 + 4^2 + (-9)^2} = \sqrt{121 + 16 + 81} = \sqrt{218} \approx 14.7648 \, \text{meters}
$$

### Step 3: Apply Coulomb's Law
$$
\vec{F} = k \frac{Q_A Q_B}{R^3} \vec{R}
$$
Substituting the known values:
$$
Q_A Q_B = (-20 \times 10^{-6} \, C)(50 \times 10^{-6} \, C) = -1 \times 10^{-9} \, \text{C}^2
$$
$$
R^3 = (14.7648)^3 \approx 3228.4 \, \text{m}^3
$$
$$
\vec{F} = \frac{8.99 \times 10^{9} \times (-1 \times 10^{-9})}{3228.4} \times (11, \, 4, \, -9)
$$
$$
\vec{F} \approx \frac{-8.99}{3228.4} \times (11, \, 4, \, -9) \approx -0.002783 \times (11, \, 4, \, -9) \, \text{N}
$$

### Step 4: Calculate the Components
$$
F_x = -0.002783 \times 11 \approx -0.0306 \, \text{N}
$$
$$
F_y = -0.002783 \times 4 \approx -0.0111 \, \text{N}
$$
$$
F_z = -0.002783 \times (-9) \approx 0.0250 \, \text{N}
$$

### Final Vector Force:
$$
\vec{F} \approx (-0.0306 \, \mathbf{i} - 0.0111 \, \mathbf{j} + 0.0250 \, \mathbf{k}) \, \text{N}
$$

### Summary:
The vector force exerted on $$ Q_A $$ by $$ Q_B $$ is approximately:
$$
\vec{F} \approx -0.0306\, \mathbf{i} - 0.0111\, \mathbf{j} + 0.0250\, \mathbf{k} \, \text{N}
$$
The vector force on $$ Q_A $$ by $$ Q_B $$ is approximately $$ -0.0306\, \mathbf{i} - 0.0111\, \mathbf{j} + 0.0250\, \mathbf{k} \, \text{N} $$.

Charge is located at and change is at free space The vector force excented on by is

Where is coulomb constant

Upstudy AI Solution

Answer

Solution

The Deep Dive

Related Questions

Latest Physics Questions

Charge is located at and change is at free space The vector force excented on by is Where is coulomb constant