To calculate the electric force between two point charges in free space, we use Coulomb's Law, which is given by:

$\vec{F} = k_e \frac{q_1 q_2}{r^2} \hat{r}$

Where:

• $k_e = 8.99 \times 10^9 \, \mathrm{N \cdot m^2 / C^2}$ is Coulomb's constant,
• $q_1 = -300 \, \mu C = -300 \times 10^{-6} \, C$,
• $q_2 = 10 \, \mu C = 10 \times 10^{-6} \, C$,
• $\vec{r} = (x_2 - x_1, y_2 - y_1, z_2 - z_1)$ is the displacement vector from charge $q_1$ to charge $q_2$,
• $r = \|\vec{r}\|$ is the magnitude of the displacement vector.

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Step 1: Displacement Vector

The coordinates of $q_1$ and $q_2$ are:

• $q_1$: $(2, 4, 5)$,
• $q_2$: $(1, 1, 3)$.

The displacement vector is: $\vec{r} = (1 - 2, 1 - 4, 3 - 5) = (-1, -3, -2).$

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Step 2: Magnitude of the Displacement Vector

The magnitude of $\vec{r}$ is: $r = \sqrt{(-1)^2 + (-3)^2 + (-2)^2} = \sqrt{1 + 9 + 4} = \sqrt{14}.$

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Step 3: Unit Vector

The unit vector $\hat{r}$ is: $\hat{r} = \frac{\vec{r}}{r} = \frac{(-1, -3, -2)}{\sqrt{14}} = \left( -\frac{1}{\sqrt{14}}, -\frac{3}{\sqrt{14}}, -\frac{2}{\sqrt{14}} \right).$

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Step 4: Force Magnitude

The magnitude of the electric force is: $|\vec{F}| = k_e \frac{|q_1 q_2|}{r^2} = (8.99 \times 10^9) \cdot \frac{|(-300 \times 10^{-6})(10 \times 10^{-6})|}{(\sqrt{14})^2}.$

$|\vec{F}| = (8.99 \times 10^9) \cdot \frac{3000 \times 10^{-12}}{14}.$

$|\vec{F}| = \frac{8.99 \times 3 \times 10^{-3}}{14} = \frac{26.97 \times 10^{-3}}{14} = 1.926 \, \mathrm{N}.$

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Step 5: Force Vector

The force vector is: $\vec{F} = |\vec{F}| \cdot \hat{r} = 1.926 \cdot \left( -\frac{1}{\sqrt{14}}, -\frac{3}{\sqrt{14}}, -\frac{2}{\sqrt{14}} \right).$

Simplifying: $\vec{F} = \left( -\frac{1.926}{\sqrt{14}}, -\frac{3 \cdot 1.926}{\sqrt{14}}, -\frac{2 \cdot 1.926}{\sqrt{14}} \right).$

Numerically: $\vec{F} = (-0.515, -1.545, -1.030) \, \mathrm{N}.$

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Final Answer:

The electric force on the charge $q_1$ is: $\vec{F} = (-0.515 \, \hat{i}, -1.545 \, \hat{j}, -1.030 \, \hat{k}) \, \mathrm{N}.$

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Would you like further clarification or another example?

5 Related Questions:

1. How would the force change if the charges were doubled?
2. What would be the force if the charges were in a medium with relative permittivity $\epsilon_r$?
3. How do you calculate the potential energy between two charges?
4. What happens to the force if the distance between the charges is halved?
5. How does the direction of the force depend on the signs of the charges?

Tip:

Always check the sign of the charges to determine if the force is attractive or repulsive.