Two like charges of the same magnitude are 8.0 mm apart. if the force of repulsion they exert upon each other is 4.0 N, what is the magnitude of each charge? (The constant of proportional for the Coulombic force is 9.0 x 10^9 N.m^2/C^2 )
Please show all work.
F = k q^2/r^2
so
q = r sqrt (F/k)
r = .008
F = 4
k = 9 * 10^9
Well, well, well, two charged buddies causing some repulsion, huh? Let's dig in and solve this chargy puzzle!
First off, we need to recall Coulomb's Law, which states that the magnitude of the electrostatic force between two charged objects is given by the formula:
F = k * |q1| * |q2| / r^2
where F is the force of repulsion, k is the constant of proportionality (9.0 x 10^9 N.m^2/C^2), |q1| and |q2| are the magnitudes of the charges, and r is the distance between the charges.
We know that F = 4.0 N and r = 8.0 mm = 0.008 m. Now we just need to rearrange the formula to solve for the magnitude of each charge!
4.0 N = (9.0 x 10^9 N.m^2/C^2) * |q1| * |q2| / (0.008 m)^2
Simplifying a bit:
(0.008 m)^2 * 4.0 N = (9.0 x 10^9 N.m^2/C^2) * |q1| * |q2|
0.000064 N.m^2 = (9.0 x 10^9 N.m^2/C^2) * |q1| * |q2|
Dividing both sides by (9.0 x 10^9 N.m^2/C^2):
0.00000000711 C^2 = |q1| * |q2|
Now, since both charges have the same magnitude, we can say that |q1| = |q2| = q. So we substitute q for both magnitudes:
0.00000000711 C^2 = q * q
Taking the square root of both sides:
0.000002665 C ≈ q
Therefore, each charge has a magnitude of approximately 0.000002665 C.
Keep in mind that this is just an approximate value. And remember, even though these charges may be small, their repulsion can still be quite shocking!
To find the magnitude of each charge, we can use Coulomb's law:
F = (k * |q1| * |q2|) / r^2
where:
F is the force of repulsion,
k is the Coulomb's constant (9.0 x 10^9 N⋅m^2/C^2),
|q1| and |q2| are the magnitudes of the charges, and
r is the distance between the charges.
Given:
F = 4.0 N,
r = 8.0 mm = 8.0 x 10^-3 m,
k = 9.0 x 10^9 N⋅m^2/C^2.
First, let's rearrange the formula to solve for the magnitude of each charge:
|q1| = (F * r^2) / (k * |q2|)
Substituting the given values into the equation:
|q1| = (4.0 N * (8.0 x 10^-3 m)^2) / (9.0 x 10^9 N⋅m^2/C^2 * |q2|)
Simplifying:
|q1| = (4.0 N * 64 x 10^-6 m^2) / (9.0 x 10^9 N⋅m^2/C^2 * |q2|)
|q1| = (0.256 x 10^-5 N⋅m^2) / (9.0 x 10^9 N⋅m^2/C^2 * |q2|)
|q1| = (2.56 x 10^-6 N⋅m^2) / (9.0 x 10^9 N⋅m^2/C^2 * |q2|)
|q1| = 2.56 / (9.0 x 10^9 C^2 * |q2|)
|q1| = 2.56 x 10^-10 / |q2|
Therefore, the magnitude of the first charge |q1| is 2.56 x 10^-10 / |q2|.
Since the problem does not provide information about the magnitude of the second charge, we cannot determine the exact value of |q1|. We can only express it in terms of |q2|.
Please provide the magnitude of |q2| to determine the magnitude of each charge.
To find the magnitude of each charge, we can use Coulomb's Law, which states that the force of repulsion between two charges is given by:
F = (k * |q1| * |q2|) / r^2
where:
- F is the force of repulsion
- k is the Coulombic constant (9.0 x 10^9 N.m^2/C^2)
- |q1| and |q2| are the magnitudes of the charges
- r is the distance between the charges
In this case, we know that the distance between the charges is 8.0 mm, which is equivalent to 0.008 m. We also know that the force of repulsion is 4.0 N.
Substituting the given values into Coulomb's Law, we have:
4.0 N = (9.0 x 10^9 N.m^2/C^2) * (|q1| * |q2|) / (0.008 m)^2
Simplifying the equation:
4.0 N = (9.0 x 10^9 N.m^2/C^2) * (|q1| * |q2|) / 6.4 x 10^-5 m^2
Now, we can solve for the magnitude of each charge by rearranging the equation:
(|q1| * |q2|) = (4.0 N * 6.4 x 10^-5 m^2) / (9.0 x 10^9 N.m^2/C^2)
(|q1| * |q2|) = 2.56 x 10^-4 N.m^2/C^2
To find the magnitude of each charge, we need to divide this value by one of the charges. Since the two charges have the same magnitude, we can write:
(|q1| * |q1|) = 2.56 x 10^-4 N.m^2/C^2
Taking the square root of both sides:
|q1| = √(2.56 x 10^-4 N.m^2/C^2)
Calculating this value:
|q1| = 5.0 x 10^-3 C
Therefore, the magnitude of each charge is 5.0 x 10^-3 C.