(a) Two point charges totaling 8.50 µC exert a repulsive force of 0.100 N on one another when separated by 0.839 m. What is the charge on each?
smallest charge (µC)
largest charge (µC)
(b) What is the charge on each if the force is attractive?
smallest charge (µC)
largest charge (µC)
a. use coulombs equation
find the product q1q2
you are given q1+q2=8.5
you should get two solutions out of this, it should be a quadratic.
To solve this problem, we can use Coulomb's law, which states that the force between two point charges is given by:
\[ F = \frac{k \cdot q_1 \cdot q_2}{r^2} \]
where F is the force, k is the electrostatic constant (9.0 x 10^9 N m^2/C^2), q1 and q2 are the charges, and r is the distance between the charges.
(a) Since the given force is repulsive, the charges must have the same sign. Let's assume the charges are q1 and q2. We can set up the following equation:
\[ \frac{k \cdot q_1 \cdot q_2}{r^2} = 0.100 \, \text{N} \]
Substituting the given values, we get:
\[ \frac{9.0 \times 10^9 \cdot q_1 \cdot q_2}{(0.839)^2} = 0.100 \]
Simplifying the equation, we have:
\[ q_1 \cdot q_2 = \frac{0.100 \times (0.839)^2}{9.0 \times 10^9} \]
Plugging the numbers in the calculator, we find:
\[ q_1 \cdot q_2 \approx 7.83 \times 10^{-12} \, \text{C}^2 \]
Since the charges have the same magnitude, we can write:
\[ q_1 = q_2 = \sqrt{7.83 \times 10^{-12}} \, \text{C} \]
Converting the charges to microcoulombs:
\[ q_1 \approx q_2 \approx 2.80 \, \mu\text{C} \]
Therefore, the smallest charge is approximately 2.80 µC, and the largest charge is also approximately 2.80 µC.
(b) If the force is attractive, the charges must have opposite signs. Again, let's assume the charges are q1 and q2. We can set up the similar equation:
\[ \frac{k \cdot q_1 \cdot q_2}{r^2} = 0.100 \, \text{N} \]
Substituting the given values, we have:
\[ \frac{9.0 \times 10^9 \cdot q_1 \cdot q_2}{(0.839)^2} = 0.100 \]
Simplifying the equation:
\[ q_1 \cdot q_2 = \frac{0.100 \times(0.839)^2}{9.0 \times 10^9} \]
Plugging the numbers in the calculator, we find:
\[ q_1 \cdot q_2 \approx 7.83 \times 10^{-12} \, \text{C}^2 \]
Since the charges have opposite signs, we can write:
\[ q_1 = -q_2 = \sqrt{7.83 \times 10^{-12}} \, \text{C} \]
Converting the charges to microcoulombs:
\[ q_1 \approx q_2 \approx 2.80 \, \mu\text{C} \]
Therefore, the smallest charge is approximately -2.80 µC, and the largest charge is also approximately 2.80 µC.
To determine the charge on each point charge, we can use Coulomb's law, which states that the force between two charged objects is directly proportional to the product of their charges and inversely proportional to the square of the distance between them. Mathematically, it can be written as:
F = k * (|q1| * |q2|) / r^2
where F is the force, k is the electrostatic constant (9 × 10^9 Nm^2/C^2), q1 and q2 are the charges, and r is the distance between the charges.
(a) For repulsive force:
Given:
F = 0.100 N
r = 0.839 m
Total charge, q1 + q2 = 8.50 µC
We need to find the charge on each point charge.
Using Coulomb's law, we can rearrange the equation to solve for the charge on each point charge:
|q1| * |q2| = (F * r^2) / k
|q1| * |q2| = (0.100 N * (0.839 m)^2) / (9 × 10^9 Nm^2/C^2)
|q1| * |q2| = 6.55 × 10^-12 C^2
Since the charges are repulsive, they have the same magnitude but opposite signs. Therefore, we can write:
q1 * q2 = -6.55 × 10^-12 C^2
To find the charge on each point charge, we can solve this equation by assuming q1 and q2 to be x and y, respectively:
xy = -6.55 × 10^-12
We need to find two numbers whose product is approximately -6.55 × 10^-12. One possible set of values is x = -5.0 × 10^-12 C and y = 1.3 × 10^-12 C.
Thus, the charge on the smallest point charge is -5.0 µC, and the charge on the largest point charge is 1.3 µC.
(b) For attractive force:
The process is similar to the previous part; the only difference is that the charges have the same sign.
Given:
F = 0.100 N
r = 0.839 m
Total charge, q1 + q2 = 8.50 µC
Using the same equation for Coulomb's law:
|q1| * |q2| = (0.100 N * (0.839 m)^2) / (9 × 10^9 Nm^2/C^2)
|q1| * |q2| = 6.55 × 10^-12 C^2
Since the charges are attractive, they have the same sign. Therefore, we can write:
q1 * q2 = 6.55 × 10^-12 C^2
Again, assuming q1 and q2 to be x and y, respectively, we can solve this equation. One possible set of values is x = y = 2.7 µC.
Thus, the charge on the smallest point charge is 2.7 µC, and the charge on the largest point charge is also 2.7 µC.