What is the magnitude if the total force exerted by the four charges

[pr_angeleyes]

Four charges of magnitude +q are placed at the corners of a square whose sides have a length d. What is the magnitude if the total force exerted by the four charges on a charge Q located a distance b along a line perpendicular to the plane of the square and equidistant from the four charges?


The result of my attempt was:

F= 4kqQb/((d^2/2)+b^2)^3/2

but I don't know if is
F= kqQb/(b^2+l^2/2)^(3/2)

 

[rude man]

Four charges of magnitude +q are placed at the corners of a square whose sides have a length d. What is the magnitude if the total force exerted by the four charges on a charge Q located a distance b along a line perpendicular to the plane of the square and equidistant from the four charges?


The result of my attempt was:

F= 4kqQb/((d^2/2)+b^2)^3/2

The above is almost right. Look again at the term d^2/2. Is it correct?

 

[Andrew Mason]

Four charges of magnitude +q are placed at the corners of a square whose sides have a length d. What is the magnitude if the total force exerted by the four charges on a charge Q located a distance b along a line perpendicular to the plane of the square and equidistant from the four charges?


The result of my attempt was:

F= 4kqQb/((d^2/2)+b^2)^3/2

but I don't know if is
F= kqQb/(b^2+l^2/2)^(3/2)

Perhaps you could explain your reasoning.

What is the l as in l^2/2?

It appears to me that F= 4kqQb/((d^2/2)+b^2)^3/2 is incorrect. Check the d^2/2 term in the denominator. What is the distance of a charge from the centre of the square?

AM

 

[pr_angeleyes]

This is what i was thinking:
F because of one charge=kqQ/(b^2+l^2/2) * cos(theta)

=kqQ/(b^2+l^2/2) *b/sqrt(b^2+l^2/2)

=kqQb/(b^2+l^2/2)^(3/2) (along the perpendicular )

as in plane parallel to the square net F=0

 

[pr_angeleyes]

explanation

Perhaps you could explain your reasoning.

What is the l as in l^2/2?

It appears to me that F= 4kqQb/((d^2/2)+b^2)^3/2 is incorrect. Check the d^2/2 term in the denominator. What is the distance of a charge from the centre of the square?

AM

This is what i was thinking:

F because of one charge=kqQ/(b^2+l^2/2) * cos(theta)

=kqQ/(b^2+l^2/2) *b/sqrt(b^2+l^2/2)

=kqQb/(b^2+l^2/2)^(3/2) (along the perpendicular )

as in plane parallel to the square net F=0

 

[pr_angeleyes]

explanation

The above is almost right. Look again at the term d^2/2. Is it correct?

this is what i was thinking
F because of one charge=kqQ/(b^2+l^2/2) * cos(theta)

=kqQ/(b^2+l^2/2) *b/sqrt(b^2+l^2/2)

=kqQb/(b^2+l^2/2)^(3/2) (along the perpendicular )

as in plane parallel to the square net F=0

 

[haruspex]

F= 4kqQb/((d^2/2)+b^2)^3/2

That looks right to me, but I see I'm outvoted by AM and rude man . Maybe I'm missing something.

 

[rude man]

That looks right to me, but I see I'm outvoted by AM and rude man . Maybe I'm missing something.

How about d^2/2 → (d/2)^2 = d^2/4 ?

 

[haruspex]

How about d^2/2 → (d/2)^2 = d^2/4 ?

No, why? As I read the OP, the point Q is distance b from the centre of the square.

 

[barryj]

I tend to agree with haruspex :-)

 

[rude man]

No, why? As I read the OP, the point Q is distance b from the centre of the square.

Curses, haruspex and barryj, you are right.

@Andrew Mason:
Locate the four charges at (d/2,d/2,0), (-d/2,d/2,0), -d/2,-d/2,0) and (d/2,-d/2,0) and the observation point at (0,0,b).

Then the distance from any charge is

sqrt[(d/2-0)^2 + (d/2-0)^2 + (0-b)^2] = sqrt{d^2/2 + b^2}.

My excuse: spatial relations was never my strong suit!

 

[Andrew Mason]

Curses, haruspex and barryj, you are right.

@Andrew Mason:
Locate the four charges at (d/2,d/2,0), (-d/2,d/2,0), -d/2,-d/2,0) and (d/2,-d/2,0) and the observation point at (0,0,b).

Then the distance from any charge is

sqrt[(d/2-0)^2 + (d/2-0)^2 + (0-b)^2] = sqrt{d^2/2 + b^2}.

My excuse: spatial relations was never my strong suit!

You are right. I was using the distance as √2d/2 and it got me a bit confused. Squared it is d^2/2. So I agree the OP's first answer was right. Sorry for any confusion!

AM