Consider three point charges arranged along the x axis. Charge q1 = -4.90 nC is located at x= 0.240 m and charge q2 = 2.30 nC is at x= -0.340 m . A positive point charge q3 is located at the origin.
-Calculate the value of q3 such that the net force on this point charge has a magnitude of 3.10 μN .
-Where along the x axis can q3 be placed and the net force on it be zero, other than the trivial answers of x=+∞ and x=−∞?
i dont think any 514 people that saw this understands!
To calculate the value of q3 such that the net force on this point charge has a magnitude of 3.10 μN, we can use Coulomb's Law. The formula for Coulomb's Law is:
F = (k * |q1 * q3|) / r^2
Where:
F is the net force
k is the electrostatic constant (8.99 * 10^9 Nm^2/C^2)
q1 and q3 are the charges
r is the distance between the charges
Step 1: Calculate the net force:
F = 3.10 μN = 3.10 * 10^-6 N
Step 2: Solve for q3:
Rearranging the formula:
q3 = (F * r^2) / (k * |q1|)
Substituting the values:
q3 = (3.10 * 10^-6 N * (0.240 m)^2) / (8.99 * 10^9 Nm^2/C^2 * |(-4.90 * 10^-9 C)|)
Calculating q3:
q3 = -5.41 nC
Therefore, the value of q3 that gives a net force of magnitude 3.10 μN is q3 = -5.41 nC.
To find where along the x-axis q3 can be placed so that the net force on it is zero (other than x = ±∞), we can set up the equation:
F_net = F_31 + F_32 = 0
Where F_31 is the force exerted on q3 by q1 and F_32 is the force exerted on q3 by q2.
Using Coulomb's Law, we have:
F_31 = (k * |q1 * q3|) / r_31^2
F_32 = (k * |q2 * q3|) / r_32^2
Since the magnitudes are equal, we can write:
(k * |q1 * q3|) / r_31^2 = (k * |q2 * q3|) / r_32^2
We know the values of q1, q2, and q3 from the given information. We need to find the distances r_31 and r_32.
r_31 = x
r_32 = x - (-0.340 m) = x + 0.340 m
Substituting the values into the equation, we have:
(k * |(-4.90 * 10^-9 C) * (-5.41 * 10^-9 C)|) / x^2 = (k * |(2.30 * 10^-9 C) * (-5.41 * 10^-9 C)|) / (x + 0.340 m)^2
Simplifying and solving for x, we can find where along the x-axis q3 can be placed so that the net force on it is zero.
To calculate the value of q3 such that the net force on the point charge has a magnitude of 3.10 μN, we can use Coulomb's Law. Coulomb's Law states that the force between two point charges is directly proportional to the product of their charges and inversely proportional to the square of the distance between them.
Let's denote the distance between q3 and q1 as d1 and the distance between q3 and q2 as d2. The net force on q3 is the vector sum of the forces due to q1 and q2.
1. Calculate the force between q3 and q1:
F1 = k * (q1 * q3) / d1^2
Where k is the Coulomb's constant (k ≈ 9 x 10^9 N·m^2/C^2).
2. Calculate the force between q3 and q2:
F2 = k * (q2 * q3) / d2^2
3. The net force on q3 is the vector sum of F1 and F2:
F_net = F1 + F2
Now, let's solve for the unknown charge q3.
4. Convert the given force magnitude to SI units:
F_net = 3.10 μN = 3.10 x 10^-6 N
5. Substituting the known values and the given force magnitude into the equation for F_net, we get:
3.10 x 10^-6 N = k * (q1 * q3) / d1^2 + k * (q2 * q3) / d2^2
6. Rearrange the equation to solve for q3:
q3 = (3.10 x 10^-6 N / k) * (d1^2 / (q1 * d2^2)) * (q2 - q1)
Now, let's find the position along the x-axis where the net force on q3 is zero.
1. For the net force on q3 to be zero, the magnitudes of F1 and F2 should be equal.
2. Set F1 = F2 and solve for the position x of q3:
k * (q1 * q3) / d1^2 = k * (q2 * q3) / d2^2
3. Rearrange the equation to solve for x:
x = (q1 * d1^2 - q2 * d2^2) / (q1 - q2)
This will give you the position along the x-axis where the net force on q3 is zero.