A charge Q1 0:1 mC is located at the origin in free space, whileQ2 0:2 mC is at A 0:8; 0:6; 0. Find the locus of points in the z 0plane at which the x-component of the force on a third positive charge iszero.
Question
A charge Q1 0:1 mC is located at the origin in free space, whileQ2 0:2 mC is at A 0:8; 0:6; 0. Find the locus of points in the z 0plane at which the x-component of the force on a third positive charge iszero.
Solution
The problem involves the concept of electric forces between charges, which is governed by Coulomb's Law. The force between two charges is given by:
F = k * |Q1*Q2| / r^2
where:
- F is the force between the charges,
- k is Coulomb's constant (approximately 8.99*10^9 N.m^2/C^2),
- Q1 and Q2 are the charges,
- r is the distance between the charges.
The x-component of the force on a charge due to another charge is given by:
Fx = F * cos(theta)
where theta is the angle between the force vector and the x-axis.
The problem asks for the locus of points in the z = 0 plane at which the x-component of the force on a third positive charge is zero. This means we want to find the points where Fx = 0, or equivalently, where cos(theta) = 0. This occurs when theta = 90 degrees or theta = 270 degrees, i.e., when the force vector is perpendicular to the x-axis.
Since the third charge is in the z = 0 plane, it must be somewhere along the line y = -0.6x + 0.48 in the xy-plane (which is the line perpendicular to the vector from Q1 to Q2 at the point A(0.8, -0.6, 0)).
Therefore, the locus of points at which the x-component of the force on the third charge is zero is the line y = -0.6x + 0.48 in the z = 0 plane.
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