A charge with a value of 3.0 x 10^-5 C is located 3.0 cm from a charge with a value of 6.0 x 10^-5 C. Determine the distance from the larger charge to the point where the total electric field is zero
To determine the distance from the larger charge to the point where the total electric field is zero, we can use the principle of superposition, which states that the electric field due to multiple charges is the vector sum of the individual electric fields produced by each charge.
1. Let's assume the point where the total electric field is zero is at a distance x from the larger charge.
2. The electric field produced by the larger charge can be calculated using Coulomb's Law:
Electric Field (E1) = k * Q1 / r^2,
where k is the Coulomb's constant (k = 9 x 10^9 Nm^2/C^2), Q1 is the charge of the larger charge (Q1 = 6.0 x 10^-5 C), and r is the distance from the larger charge to the point.
Thus, the electric field produced by the larger charge can be written as:
E1 = (9 x 10^9 Nm^2/C^2) * (6.0 x 10^-5 C) / (x)^2
3. The electric field produced by the smaller charge, located at a distance 3.0 cm (0.03 m) from the larger charge, can be calculated in a similar way:
Electric Field (E2) = k * Q2 / r^2,
where Q2 is the charge of the smaller charge (Q2 = 3.0 x 10^-5 C), and r is the distance from the smaller charge to the point.
Thus, the electric field produced by the smaller charge can be written as:
E2 = (9 x 10^9 Nm^2/C^2) * (3.0 x 10^-5 C) / (0.03 m)^2
4. Since the total electric field at the point where it is zero is the vector sum of E1 and E2, it must be equal to zero. Therefore, we can write the equation:
E1 + E2 = 0
(9 x 10^9 Nm^2/C^2) * (6.0 x 10^-5 C) / (x)^2 + (9 x 10^9 Nm^2/C^2) * (3.0 x 10^-5 C) / (0.03 m)^2 = 0
5. Now we can solve this equation for x.
Rearranging the equation:
(6.0 x 10^-5 C) / x^2 + (3.0 x 10^-5 C) / (0.03 m)^2 = 0
(6.0 x 10^-5 C) / x^2 + (3.0 x 10^-5 C) / (0.03)^2 = 0
Multiplying through by x^2:
(6.0 x 10^-5 C) + (3.0 x 10^-5 C) * (x^2) / (0.03)^2 = 0
(6.0 x 10^-5 C) + (3.0 / 9 x 10^-5) * x^2 = 0
6.0 + (1 / 3) x^2 = 0
1 / 3 x^2 = -6.0
x^2 = -18
Since we cannot take the square root of a negative number, there is no real solution for x. This means that there is no point where the total electric field is zero.
Therefore, there is no distance from the larger charge to a point where the total electric field is zero.