Charges q1 and q2 lie on the x-axis at points x= -a and x= +a respectively a) how must must q1 and q2 be related for the net force +Q placed at x= +a/2 to be zero?
zero, then the forces are equal and opposite.
q1*q/(3a/2)^2=q2*q/(a/2)^2
q1=9 q2
another way to look at this is q1 is 3 times the distance from q as is q2
To find out how q1 and q2 must be related for the net force to be zero at the point x = +a/2, we can use Coulomb's law, which states that the electrical 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 derive the expression for the net force using Coulomb's law:
The electrical force between two charges q1 and q2 is given by:
F = k * q1 * q2 / r^2
Where:
- F is the force between the charges,
- k is the electrostatic constant (approximately 9 × 10^9 Nm^2/C^2),
- q1 and q2 are the magnitudes of the charges,
- r is the distance between the charges.
Since the net force at x = +a/2 is zero, the forces exerted by charges q1 and q2 must be equal in magnitude but opposite in direction. Hence, the force exerted by q1 on +Q must be equal in magnitude but opposite in direction to the force exerted by q2 on +Q.
Let's denote the force exerted by q1 on +Q as F1, and the force exerted by q2 on +Q as F2.
At x = +a/2, the distance between q1 and +Q is a/2, and the distance between q2 and +Q is a/2:
F1 = k * q1 * Q / (a/2)^2
F2 = k * q2 * Q / (a/2)^2
Since we want the net force to be zero, F1 must be equal in magnitude but opposite in direction to F2:
F1 = -F2
k * q1 * Q / (a/2)^2 = -k * q2 * Q / (a/2)^2
Dividing both sides by Q and multiplying by (a/2)^2:
k * q1 = -k * q2
Simplifying:
q1 = -q2
Therefore, the charges q1 and q2 must have opposite signs and be equal in magnitude for the net force to be zero at x = +a/2.
To find the relationship between q1 and q2 for the net force to be zero at x = +a/2, we need to consider the forces acting on the +Q charge.
The force between two charges is given by Coulomb's law:
F = k * ( |q1*q2| / r^2 )
where F is the force, k is the electrostatic constant (k = 8.99 x 10^9 N m^2/C^2), q1 and q2 are the charges, and r is the distance between the charges.
Since the net force should be zero, the forces exerted by q1 and q2 on the +Q charge should cancel each other out. We can write this condition as:
F1 + F2 = 0
Using Coulomb's law, we can rewrite this equation as:
k * ( |q1*q2| / r1^2 ) + k * ( |q1*q2| / r2^2 ) = 0
Substituting the given values:
- r1 = a/2 (distance between q1 and +Q)
- r2 = 3a/2 (distance between q2 and +Q)
The equation becomes:
k * ( |q1*q2| / (a/2)^2 ) + k * ( |q1*q2| / (3a/2)^2 ) = 0
Simplifying further:
4 * ( |q1*q2| / a^2 ) + (1/9) * ( |q1*q2| / a^2 ) = 0
Combining the terms and rearranging:
(36 + 1) * ( |q1*q2| / a^2 ) = 0
37 * ( |q1*q2| / a^2 ) = 0
Since the product of charges (|q1*q2|) cannot be zero unless either q1 or q2 is zero, we can conclude that q1 and q2 must have opposite signs for the net force to be zero at x = +a/2. Therefore, q1 and q2 must be related as opposite charges.