## One-Dimension Multiple Charge Problems

So far we have seen that electrical forces can be calculated between two point charges.

What happens when there are more than two charges involved? The resulting force on a charged particle will be equal to the vector sum of all of the electrostatic forces acting on that particle. This is known as the **Principle of Superposition**.

Problem #1: Three charges lie on the x-axis. A is -4.0 nC at x = 0.50 m, B is 5.0 nC at x=0, and C is 3.0 nC at x = – 0.80 m. Find the magnitude and direction of the electrostatic force on the 5.0 nC charge, charge B.

Solution: The 5.0 nC charge is attracted to the -4 nC charge and repelled from the 3.0 nC charge. Therefore both forces will be directed to the left. In force notation, F_{BA} <– (left) and F_{BC} <– (left). (Remember F_{BA} means the force that Charge B exerts on Charge A and F_{BC} means the force that Charge B exerts on Charge C.)

Calculate the individual forces using Coulombs Law:

F_{BA} = __(9 x 10 ^{9} Nm^{2}/C^{2})(5.0 x 10^{-9} C)(-4.0 x 10^{-9} C)__

(0.50 m)^{2}

F_{BA} = – 7.2 x10^{-7} N, left (negative indicates attraction, not magnitude)

F_{BC} = __(9 x 10 ^{9} Nm^{2}/C^{2})(5.0 x 10^{-9} C)(3.0 x 10^{-9} C)__

(0.80 m)^{2}

F_{BC} = +2.1 x 10^{-7} N, left (positive indicates repulsion, not magnitude)

F_{B} = F_{BA} + F_{BC} = 9.3 x 10^{-7} N, left

Problem #2: Using the same charges as Problem #1, find the net force on Charge C.

F_{CA} = __(9 x 10 ^{9} Nm^{2}/C^{2})(3.0 x 10^{-9} C)(-4.0 x 10^{-9} C),__ LEFT (attracted)

(1.30 m)^{2}

F_{CA} = -6.4 x 10^{-8} N, left

F_{CB} = __(9 x 10 ^{9} Nm^{2}/C^{2})(3.0 x 10^{-9} C)(5.0 x 10^{-9} C),__ right (repelled)

(0.80 m)^{2}

F_{CB} = +2.1 x 10^{-7} N, right

(NOTICE: F_{BC} = -F_{CB})

**Since the two vector quantities are opposite in direction, we take the difference between their magnitudes (ignore the signs) and assign the direction of the greater individual magnitude.**

F_{C} = |F_{CA –} F_{CB}|

F_{C} = 1.5 x 10^{-7} N, right

### Try the following problems:

1) How many electrons are contained in 1 C of charge? What is the mass of the electrons in 1 C of charge?

2) If two equal charges, each of 1 C, were separated in air by a distance of 1 km, what would be the force between them?

3) Determine the force between two free electrons spaced 1 angstrom (10^{-10} m) apart.

4) What is the force of repulsion between two argon nuclei that are separated by 1 nm? The charge on an argon nucleus is +18e.

(source)

Note for Lee:

Another resource for multiple charges, however, it’s missing scientific notation marks such as .5 x 10^8,