Two electrical charges, one a positive charge A of magnitude a and the other a negative charge B of magnitude b, are loacted a distance c apart. A positively charged Particle P is situated on the line between A and B. Find where P should be put on so that the pull away from A towards B is minimal. Here assumee that the force from each charge is proportional to the strength of the source and inversely proportional to the square of the distance from the source.
Let x be the distance from P to A, and so c-x is the distance from P to B.
The net force acting upon P, while it is between A and B, is
F = k q [a/x^2 + b/(c-x^)2]
a and b are positive charge magnitud3es and q is the charge of P. k is a Boltzman contant. The force is a minimum when dF/dx = 0
This leads to
-2 a/x^3 + 2 b/(c-x)^-3
(b/a) = [(c-x)/x]^3 = [(c/x)-1]^3
This can be solved for x/c in terms of b/a
To find where P should be placed on the line between A and B such that the pull away from A towards B is minimal, we need to minimize the net force acting on P. The net force can be calculated using Coulomb's law, which states that the force between two charges is proportional to the product of their magnitudes and inversely proportional to the square of the distance between them.
Let's denote the distance from P to A as x, and the distance from P to B as (c - x), where c is the distance between charges A and B.
The net force acting on P, while it is between A and B, can be expressed as:
F = kq [a/x^2 + b/(c - x)^2]
In this equation, a and b are the magnitudes of charges A and B respectively, q is the charge of P, and k is a constant.
To find where the force is minimal, we need to find the value of x that makes the derivative of the force equation equal to zero. Taking the derivative of the force equation with respect to x:
dF/dx = -2a/x^3 + 2b/(c - x)^3
Setting dF/dx equal to zero and simplifying:
-2a/x^3 + 2b/(c - x)^3 = 0
This can be further simplified by dividing both sides of the equation by -2:
a/x^3 - b/(c - x)^3 = 0
Now we can rearrange the equation to solve for the ratio of (c - x) to x:
(b/a) = [(c - x)/x]^3
Taking the cube root of both sides to eliminate the exponent:
( (c - x)/x ) = (b/a)^(1/3)
Now we can solve this equation to find the value of x/c in terms of b/a. By solving for x/c, we can locate the position of P on the line between A and B that minimizes the net force of attraction towards B.