Question

Electrostatics - Force on a Concentrated Charge when Placed in a Group of Point Charges

Answer 1

When a concentrated charge is placed in the vicinity of a group of other point charges, it experiences a net electrostatic force due to the interaction with those charges. The total electrostatic force on the concentrated charge is the vector sum of the individual forces exerted by each of the other point charges.

The electrostatic force between two point charges q1 and q2, separated by a distance r, is given by Coulomb's Law:


=


1

2

2
,
F=k
r
2
q
1
    ​

q
2
    ​

    ​

,

where:


F is the magnitude of the electrostatic force between the charges,

k is Coulomb's constant (
8.988
×
1
0
9
 
N m
2
/
C
2
8.988×10
9
N m
2
/C
2
 in SI units),

1
q
1
    ​

 and

2
q
2
    ​

 are the magnitudes of the charges,

r is the distance between the charges.

To find the net force on a concentrated charge

Q placed among a group of other point charges, you need to calculate the individual forces between

Q and each of the other charges, and then vectorially sum these forces to obtain the total force.

Let's say you have

n point charges


q
i
    ​

 located at positions


r
i
    ​

 in three-dimensional space. The net force

net
F
net
    ​

 on the concentrated charge

Q due to these charges is given by:


net
=
∑

=
1



,
F
net
    ​

=∑
i=1
n
    ​

F
i
    ​

,

where


F
i
    ​

 is the force exerted by the

i-th charge on

Q and is calculated using Coulomb's Law as described above.

Each individual force


F
i
    ​

 can be broken down into its Cartesian components as follows:



=




+




+




,
F
i
    ​

=F
xi
    ​

i+F
yi
    ​

j+F
zi
    ​

k,

where



F
xi
    ​

,



F
yi
    ​

, and



F
zi
    ​

 are the components of


F
i
    ​

 along the

x,

y, and

z axes, respectively.

Finally, to find the net force

net
F
net
    ​

 acting on the concentrated charge

Q, you add up the individual force components:


net
=
∑

=
1

(




+




+




)
.
F
net
    ​

=∑
i=1
n
    ​

(F
xi
    ​

i+F
yi
    ​

j+F
zi
    ​

k).

This will give you the total force vector acting on the concentrated charge

Q due to the group of point charges.

Keep in mind that this is a simplified explanation, and in more complex scenarios, you might need to consider vector addition and subtraction carefully to accurately determine the net force. Additionally, the charges and distances should be in appropriate units to ensure consistent calculations.