Math Problem Statement

Given two charges q1=5 μC and q2=−5 μC are separated by a distance r in a vacuum. If the magnitude of the force between them is 1 N, what is the separation distance r? a) 1.5 m b) 0.1 m c) 0.3 m d) 0.5 m

Solution

To find the separation distance rr between the charges q1q_1 and q2q_2, we can use Coulomb's Law, which states that the force FF between two charges is given by:

F=kq1q2r2F = k \frac{|q_1 q_2|}{r^2}

where:

  • FF is the magnitude of the force between the charges,
  • kk is Coulomb's constant (8.99×109Nm2/C28.99 \times 10^9 \, \text{N} \cdot \text{m}^2 / \text{C}^2),
  • q1q_1 and q2q_2 are the magnitudes of the charges,
  • rr is the separation distance between the charges.

Given:

  • F=1NF = 1 \, \text{N},
  • q1=5μC=5×106Cq_1 = 5 \, \mu\text{C} = 5 \times 10^{-6} \, \text{C},
  • q2=5μC=5×106Cq_2 = -5 \, \mu\text{C} = -5 \times 10^{-6} \, \text{C}.

Let's solve for rr:

1=8.99×109(5×106)(5×106)r21 = 8.99 \times 10^9 \cdot \frac{(5 \times 10^{-6}) \cdot (5 \times 10^{-6})}{r^2}

First, simplify the numerator:

(5×106)(5×106)=25×1012C2(5 \times 10^{-6}) \cdot (5 \times 10^{-6}) = 25 \times 10^{-12} \, \text{C}^2

Now substitute back into the equation:

1=8.99×10925×1012r21 = 8.99 \times 10^9 \cdot \frac{25 \times 10^{-12}}{r^2}

1=8.99×10925×1012r21 = \frac{8.99 \times 10^9 \cdot 25 \times 10^{-12}}{r^2}

1=224.75×103r21 = \frac{224.75 \times 10^{-3}}{r^2}

1=0.22475r21 = \frac{0.22475}{r^2}

Now, solve for r2r^2:

r2=0.22475r^2 = 0.22475

r=0.22475r = \sqrt{0.22475}

r0.474mr \approx 0.474 \, \text{m}

Comparing this to the given options:

a) 1.5 m
b) 0.1 m
c) 0.3 m
d) 0.5 m

The closest answer is d)0.5md) 0.5 \, \text{m}.

Thus, the separation distance rr is approximately 0.5m0.5 \, \text{m}.

Would you like more details or have any questions?

Here are five more questions you might want to ask:

  1. How do you derive Coulomb's Law?
  2. What are the units of Coulomb's constant kk?
  3. Can you explain how to convert microcoulombs to coulombs?
  4. What are some applications of Coulomb's Law in real life?
  5. How does the force between charges change if one of the charges is doubled?

Tip: Always ensure that the units are consistent when using formulas in physics to avoid calculation errors.

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Math Problem Analysis

Mathematical Concepts

Electric Charge
Coulomb's Law

Formulas

Coulomb's Law: F = k * |q1 * q2| / r^2

Theorems

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Suitable Grade Level

Grades 11-12