Suppose the charge q2 in the figure below can be moved left or right along the line connecting the charges q1 and q3. Given that q = +15 µC, find the distance from q1 where q2 experiences a net electrostatic force of zero? (The charges q1 and q3 are separated by a fixed distance of 36 cm.)
magnitude in cm ?
direction?
It matters what q1 and q3 are.
To find the distance from charge q1 where q2 experiences a net electrostatic force of zero, we can use Coulomb's Law.
Coulomb's Law states that the magnitude of the electrostatic force between two charged objects is given by:
F = (k * q1 * q2) / r^2
Where:
F is the magnitude of the electrostatic force,
k is the electrostatic constant (9 x 10^9 Nm^2/C^2),
q1 and q2 are the charges of the objects,
and r is the distance between the centers of the charges.
In this case, we have q1 = q3 = q = +15 µC, and a fixed distance between q1 and q3 of 36 cm. We'll assume q2 is initially at a distance x from q1.
To find at what distance x from q1 q2 experiences a net electrostatic force of zero, we'll set the electrostatic forces on q2 due to q1 and q3 equal to each other and solve for x.
The electrostatic force on q2 due to q1 is given by:
F1 = (k * q1 * q2) / d1^2
The electrostatic force on q2 due to q3 is given by:
F3 = (k * q3 * q2) / d2^2
Since the forces are equal, we’ll set them equal to each other:
F1 = F3
(k * q1 * q2) / d1^2 = (k * q3 * q2) / d2^2
Since q1 = q3 = q, and d1 = x (distance from q1 to q2), and d2 = 36 cm - x (distance from q3 to q2), we can substitute these values:
(k * q * q2) / x^2 = (k * q * q2) / (36 cm - x)^2
Simplifying further, we can cancel out the k * q * q2 terms:
x^2 = (36 cm - x)^2
Expanding the right side of the equation:
x^2 = 1296 cm^2 - 72 cm * x + x^2
Rearranging the terms:
72 cm * x = 1296 cm^2
Dividing both sides by 72 cm:
x = 18 cm
So, the distance from q1 where q2 experiences a net electrostatic force of zero is 18 cm.
In terms of direction, since q2 can be moved left or right along the line connecting q1 and q3, the direction would depend on the starting position of q2. If q2 is initially to the left of q1, then the direction would be towards q3 (right). If q2 is initially to the right of q1, then the direction would be towards q1 (left).
To find the distance from q1 where q2 experiences a net electrostatic force of zero, we can use the principle of Coulomb's Law, which states that the electrostatic force between two charged objects is directly proportional to the product of their charges and inversely proportional to the square of the distance between them.
Let's assign some values to the variables:
q1 = +15 µC (given)
q3 = ? (unknown)
q2 = ? (unknown)
distance between q1 and q3 = 36 cm
Since the net electrostatic force on q2 should be zero, the force exerted by q1 on q2 (feq1) should be equal in magnitude and opposite in direction to the force exerted by q3 on q2 (feq3).
Using Coulomb's Law, the force between q1 and q2 (feq1) can be calculated as:
feq1 = (k * |q1| * |q2|) / (r^2)
Similarly, the force between q3 and q2 (feq3) can be calculated as:
feq3 = (k * |q3| * |q2|) / (r^2)
Since feq1 and feq3 have equal magnitudes and opposite directions, we can set them equal to each other:
feq1 = feq3
(k * |q1| * |q2|) / (r^2) = (k * |q3| * |q2|) / (r^2)
Since we are looking for the magnitude and direction of the distance, we can cancel out the r^2 term:
k * |q1| * |q2| = k * |q3| * |q2|
Now, substituting the known values:
k * (15 µC) * |q2| = k * |q3| * |q2|
Since we are looking for the distance at which q2 experiences zero net force, the magnitude of q2 must equal zero. Therefore, the equation becomes:
k * (15 µC) * 0 = k * |q3| * 0
The right-hand side of the equation is always zero, so we can conclude that the distance from q1 where q2 experiences zero net force is independent of q2. In other words, it does not matter where q2 is along the line connecting q1 and q3; the net force on q2 will always be zero.
Hence, the distance from q1 where q2 experiences a net electrostatic force of zero is not dependent on the magnitude or direction of q2. It remains constant regardless of the value of q2.