
Red H.
asked 06/22/17Find the distance of 4th Q ?
There are given 3 source charges Q1= Q, Q2=Q and Q3= 2Q. Q3 is on top of triangle and Q1 and Q2 on bottom. The distance of q1=q2=q3= X
where to place 4th Q in way electric field equals to 0 ?
where to place 4th Q in way electric field equals to 0 ?
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1 Expert Answer

Arturo O. answered 06/22/17
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I assume that x is the distance from the center of the equilateral triangle to each corner of the equilateral triangle, and that all the charges are positive. Note that along a vertical line through 2Q bisecting the base of the triangle, the horizontal component of the field is zero, by symmetry. Place a 4th charge Q along that vertical line at a distance b below the center of the triangle. The horizontal component of the field will still be zero. Find the vertical component of the field, set it equal to zero, and solve for b.
Consider a positive test charge at the center. The 2Q on top will push it down. The Q's at the lower left and right corners will push it up. The 4th Q below the center of the triangle will also push it up.
The 2Q charge generates a vertical field component of -k(2Q)/x2 at the center (down). The other 2 corner charges each generate a vertical field component of (kQ/x2)sin30° = (1/2)kQ/x2 (up). The 4th Q generates a vertical field component of kQ/b2 (up).
Ey = -k(2Q)/x2 + (1/2)kQ/x2 + (1/2)kQ/x2 + kQ/b2 = 0
-2/x2 + 1/x2 + 1/b2 = 0
1/b2 - 1/x2 = 0
b = x
The 4th Q will be at a distance x below the center of the triangle, which puts it a little below the base of the triangle.
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Arturo O.
06/22/17