Question

Three positive charges A, B, and C, and a negative charge D are placed in a line as shown in the diagram. All four charges are of equal magnitude. The distances between A and B, B and C, and C and D are equal.

a. Which charge experiences the greatest net force? Which charge experiences the smallest net force?
b. Find the ratio of the greatest to the smallest net force.

Answer 1

a. charge C experiences the greatest net force, and charge B receives the smallest net force

b. ratio=9

Step-by-step explanation:

Electrostatic Force

Two point-charges
`q_1` and
`q_2` separated a distance d will exert a force on each other of a magnitude given by the Coulomb's formula

`\displaystyle F=(k\ q_1\ q_2)/(r^2)`

Where k is the proportional constant of value

`k=9*10^9\ N.m^2/c^2`

The diagram provided in the question shows four identical charges (let's assume their value is Q) separated by identical distance (of value d). The force between the charges next to others is

`\displaystyle F_1=(k\ Q\ Q)/(d^2)`

`\displaystyle F_1=(k\ Q^2)/(d^2)`

The force between charges separated 2d is

`\displaystyle F_2=(k\ Q^2)/((2d)^2)`

`\displaystyle F_2=(k\ Q^2)/(4d^2)`

And the force between the charges A and D is

`\displaystyle F_3=(k\ Q^2)/((3d)^2)`

`\displaystyle F_3=(k\ Q^2)/(9d^2)`

Now, let's analyze each charge and the force applied to them by the others

Let's recall equally signed charges repel each other and differently signed charges attrach each other

Charge A. It receives force to the left from B and C and to the right from D

`\displaystyle F_A=-F_1-F_2+F_3=-(k\ Q^2)/(d^2)-(k\ Q^2)/(4d^2)+(k\ Q^2)/(9d^2)`

`\displaystyle F_A=(k\ Q^2)/(d^2)(-1-(1)/(4)+(1)/(9))`

`\displaystyle F_A=-(41)/(36)F_1`

Charge B. It receives force to the right from A and D and to the left from C

`\displaystyle F_B=F_1-F_1+F_2=(k\ Q^2)/(d^2)-(k\ Q^2)/(d^2)+(k\ Q^2)/(4d^2)`

`\displaystyle F_B=(1)/(4)F_1`

Charge C. It receives forces to the right from all charges.

`\displaystyle F_C=F_2+F_1+F_1=(k\ Q^2)/(4d^2)+(k\ Q^2)/(d^2)+(k\ Q^2)/(d^2)`

`\displaystyle F_C=(9)/(4)F_1`

Charge D. It receives forces to the left from all charges

`\displaystyle F_D=-F_3-F_2-F_1=-(k\ Q^2)/(9d^2)-(k\ Q^2)/(4d^2)-(k\ Q^2)/(d^2)`

`\displaystyle F_D=-(49)/(36)F_1`

Comparing the magnitudes of each force is just a matter of computing the fractions

`\displaystyle (41)/(36)=1.13,\ (1)/(4)=0.25,\ (9)/(4)=2.25,\ (49)/(36)=1.36`

a.

We can see the charge C experiences the greatest net force, and charge B receives the smallest net force

b.

The ratio of the greatest to the smallest net force is

`\displaystyle ((9)/(4))/((1)/(4))=9`

The greatest force is 9 times the smallest net force