June 2020 Test
Three charges lie along the x-axis.
One positive. charge q1=15micro coulomb is at x =2.0m and another positive charge q2 =6.0 micro coulomb is at the origin. At what point on the x-axis must a negative charge q3 be placed so that the resultant force on it be zero?
To find the point on the x-axis where a negative charge q3 should be placed in order for the resultant force on it to be zero, we need to consider the forces acting on q3 due to q1 and q2.
Let's denote the position of q3 as x.
The electric force between two charges can be calculated using Coulomb's Law:
F = k * |q1 * q3| / r1^2 (1)
F = k * |q2 * q3| / r2^2 (2)
where F is the force, k is the electrostatic constant (9 x 10^9 N m^2/C^2), q1 and q2 are the magnitudes of charges, and r1 and r2 are the distances between the respective charges and q3.
Since we want the resultant force on q3 to be zero, we can set the forces calculated using equations (1) and (2) to be equal in magnitude:
k * |q1 * q3| / r1^2 = k * |q2 * q3| / r2^2
Now, let's substitute the given values:
(9 x 10^9 N m^2/C^2) * |(15 x 10^-6 C) * q3| / (2.0 m)^2 = (9 x 10^9 N m^2/C^2) * |(6 x 10^-6 C) * q3| / (x)^2
Simplifying the equation:
15000 / 4 = 6000 / x^2
Cross multiplying:
15000 * x^2 = 24000
Dividing both sides by 15000:
x^2 = 24000 / 15000 = 8 / 5
Taking the square root of both sides:
x = sqrt(8 / 5) = sqrt(8) / sqrt(5) = 2.83 / sqrt(5)
Therefore, the point on the x-axis where a negative charge q3 should be placed so that the resultant force on it is zero is approximately "2.83 / sqrt(5) meters" from the origin.
To find the position where a negative charge q3 should be placed on the x-axis such that the resultant force on it is zero, we can use the principle of superposition and Coulomb's Law.
Coulomb's Law states that the force between two charged particles is directly proportional to the product of their charges and inversely proportional to the square of the distance between them. Mathematically, it can be expressed as:
F = k * (q1 * q3) / r1^2 + k * (q2 * q3) / r2^2
Where:
F is the resultant force on q3,
k is Coulomb's constant (k = 9 * 10^9 Nm^2/C^2),
q1 and q2 are the charges of q1 and q2 respectively,
q3 is the charge of q3,
r1 is the distance between q1 and q3,
r2 is the distance between q2 and q3.
To find the position of q3 where the resultant force is zero, we need to equate the magnitudes and opposite directions of the forces generated by q1 and q2 on q3. Since one charge is positive and the other charge is negative, they will naturally repel each other.
So, the force on q3 due to q1 can be written as:
F1 = k * (q1 * q3) / r1^2
And the force on q3 due to q2 can be written as:
F2 = k * (q2 * q3) / r2^2
Since the forces are equal in magnitude but opposite in direction, we can set F1 equal to F2 and rearrange the equation as follows:
F1 = F2
k * (q1 * q3) / r1^2 = k * (q2 * q3) / r2^2
Cancelling out the k from both sides and rearranging the equation, we get:
(q1 * q3) / r1^2 = (q2 * q3) / r2^2
Now, we can substitute the given values into the equation:
(15 * 10^(-6) C * q3) / (2.0 m)^2 = (6 * 10^(-6) C * q3) / (0.0 m)^2
Simplifying the equation, we get:
225 * 10^(-6) / 4 = 6 * 10^(-6) / 0
Since the denominator of the right side is zero, we cannot divide by zero. Therefore, there is no specific position on the x-axis where a negative charge q3 can be placed to make the resultant force on it zero.
electrostatic force is an inverse-square relation
6.0 / d^2 = 15 /(2.0 - d)^2
solve for d
15 d^2 = 24 - 24 d + 6 d^2
9 d^2 + 24 d - 24 = 0 ... 3 d^2 + 8 d - 8 = 0