two point charges, Q1=-25microcoulomb and Q2=+50 microcoulomb, are separated by a distance of 12 cm. the electric field at the point P is zero. how far from Q1 is P?
@__________@_______________________@
P -25 +50
Well, E from q1 goes left, and E from Q2 goes left at P. So how exactly is it ever going to be zero between the charges if they are in the same direction?
29
To find the distance from Q1 to point P where the electric field is zero, we can use the concept of electric field due to a point charge.
The electric field due to a point charge can be calculated using the formula:
E = k * Q / r^2
where E is the electric field, k is the electrostatic constant (k = 9 * 10^9 Nm^2/C^2), Q is the charge, and r is the distance from the charge.
Since the electric field at point P is zero, we can set up the equation for the electric fields due to Q1 and Q2:
E1 = k * Q1 / r1^2
E2 = k * Q2 / r2^2
Since E1 + E2 = 0, we have:
k * Q1 / r1^2 + k * Q2 / r2^2 = 0
Substituting the given values into the equation:
(9 * 10^9)(-25 * 10^(-6)) / r1^2 + (9 * 10^9)(50 * 10^(-6)) / r2^2 = 0
Simplifying further:
(-225 * 10^3) / r1^2 + (450 * 10^3) / r2^2 = 0
Since we know that Q1 and Q2 are separated by a distance of 12 cm (or 0.12 m), we can substitute r2 = 0.12 m - r1 into the equation:
(-225 * 10^3) / r1^2 + (450 * 10^3) / (0.12 - r1)^2 = 0
Multiplying through by r1^2 * (0.12 - r1)^2:
(-225 * 10^3)(0.12-r1)^2 + (450 * 10^3)r1^2 = 0
Expanding and rearranging the equation:
(0.012 - 0.24r1 + r1^2) * (-225 * 10^3) + (450 * 10^3)r1^2 = 0
Simplifying and rearranging further:
(450 * 10^3)r1^2 - (45 * 10^3)r1 + (2.7 * 10^3) = 0
Solving this quadratic equation using the quadratic formula:
r1 = (-b ± √(b^2 - 4ac)) / (2a)
where a = (450 * 10^3), b = (-45 * 10^3), and c = (2.7 * 10^3).
Calculating the values:
r1 = [(-(-45 * 10^3)) ± √((-45 * 10^3)^2 - 4(450 * 10^3)(2.7 * 10^3))] / (2(450 * 10^3))
Simplifying further:
r1 = [45 * 10^3 ± √(2025 * 10^6 - 4 * (450 * 10^3)(2.7 * 10^3))] / (900 * 10^3)
Calculating the values inside the square root:
r1 = [45 * 10^3 ± √(2025 * 10^6 - 4 * (450 * 10^3)(2.7 * 10^3))] / (900 * 10^3)
r1 = [45 * 10^3 ± √(2025 * 10^6 - 2430 * 10^6)] / (900 * 10^3)
r1 = [45 * 10^3 ± √(-4275 * 10^6)] / (900 * 10^3)
Since the term inside the square root is negative, it means that there is no real solution for r1. Therefore, there is no distance from Q1 where the electric field is zero.
To find the distance from Q1 to point P where the electric field is zero, we can use the concept of electric fields created by point charges.
The electric field at a point due to a point charge is given by the formula:
E = k * (Q / r^2)
where E is the electric field, k is the Coulomb's constant (8.99 x 10^9 Nm^2/C^2), Q is the charge of the point charge, and r is the distance between the point charge and the point where the electric field is being measured.
In this case, at point P, the electric field is zero. Therefore, the equation becomes:
0 = k * (Q1 / r^2)
Simplifying the equation, we get:
Q1 / r^2 = 0
Since Q1 is a non-zero value, the only way for this equation to hold true is if r^2 is infinite or r is infinite. However, in a physical scenario, we cannot have an infinite distance between the charges.
Therefore, it is not possible for the electric field at point P to be zero.
It is important to double-check the given information or verify if there might be any mistake in the problem statement.