Consider two charges, where one (+2.10 nC) is at the origin and the other (-4.41 nC) is at the position x = 2.45 cm.
Find the x-coordinate where a proton would experience zero net force.
We do not care if it is a proton or an electron or a uranium nucleus at point x, call the charge at x Q
we do not care if k is 9 * 10^9 or just k
we do not care if they are nC or C, just their ratio matters :)
we do not care if it is meters or centimeters or miles
only ratios matter here
we want the point where the electric field due to the two charges is zero
k * 2.1* Q / x^2 - k*4.41 * Q/ (x-2.45)^2 = 0
2.1/x^2 = 4.41/(x-2.45)^2 for answer in centimeters
https://www.mathsisfun.com/quadratic-equation-solver.html
To find the x-coordinate where a proton would experience zero net force, we need to calculate the electric forces exerted on the proton by each of the charges and find the position where these forces cancel each other out.
First, let's calculate the electric force exerted on the proton by the charge at the origin (q1= +2.10 nC). The electric force between the two charges can be calculated using Coulomb's law:
F1 = k * |q1 * q2| / r^2
where:
- F1 is the electric force exerted on the proton by the charge at the origin,
- k is the electrostatic constant (k ≈ 9.0 x 10^9 N m^2/C^2),
- q1 is the charge at the origin (+2.10 nC),
- q2 is the charge of the proton (+1.60 x 10^-19 C),
- r is the distance between the two charges (which is the x-coordinate we want to find).
Now, let's calculate the electric force exerted on the proton by the charge at the position x = 2.45 cm (q2= -4.41 nC). Again, we use Coulomb's law:
F2 = k * |q1 * q2| / r^2
where:
- F2 is the electric force exerted on the proton by the charge at the position x = 2.45 cm,
- k is the electrostatic constant,
- q1 is the charge at the origin,
- q2 is the charge (-4.41 nC),
- r is the distance between the two charges (which is the x-coordinate we want to find).
Since we want the net force on the proton to be zero, the magnitudes of the forces must be equal:
|F1| = |F2|
Now we can set up the equations using the given values and solve for the x-coordinate:
k * |q1 * q2| / r1^2 = k * |q1 * q2| / r2^2
Simplifying and rearranging:
r2^2 * r1^2 = r1^2 * r2^2
Now we can solve for r2:
r2 = sqrt(r1^2 * r2^2)
Substituting the given values:
r2 = sqrt[(0.0245)^2 * (0)^2] (since r1= 0, as the first charge is at the origin)
This simplifies to:
r2 = sqrt(0)
Therefore, r2 = 0.
Hence, the x-coordinate where a proton would experience zero net force is at x = 0 cm (at the origin).