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 need to consider the electric fields created by both charges.
We can use Coulomb's Law to find the electric field due to each charge at a distance r:
Electric field (E) = (1 / (4πε₀)) * (q / r²)
where ε₀ is the permittivity of free space, q is the charge, and r is the distance.
Let's calculate the electric field due to the smaller charge (3.0 x 10^-5 C) at a distance of 3.0 cm:
Electric field (E1) = (1 / (4πε₀)) * ((3.0 x 10^-5 C) / (0.03 m)²)
Now, let's calculate the electric field due to the larger charge (6.0 x 10^-5 C) at an unknown distance (let's call it x):
Electric field (E2) = (1 / (4πε₀)) * ((6.0 x 10^-5 C) / (x)²)
Since we want the total electric field to be zero, the electric fields due to both charges must cancel out:
E1 + E2 = 0
To solve this equation, we substitute the values of E1 and E2 into the equation:
(1 / (4πε₀)) * ((3.0 x 10^-5 C) / (0.03 m)²) + (1 / (4πε₀)) * ((6.0 x 10^-5 C) / (x)²) = 0
Now, we can simplify the equation:
(3.0 x 10^-5 C) / (0.09 m²) + (6.0 x 10^-5 C) / (x)² = 0
Multiplying through by (0.09 m²)(x)², we get:
(3.0 x 10^-5 C)(x)² + (6.0 x 10^-5 C)(0.09 m²) = 0
Rearranging the equation, we have:
(3.0 x 10^-5 C)(x)² = -(6.0 x 10^-5 C)(0.09 m²)
Dividing both sides by (3.0 x 10^-5 C), we get:
(x)² = -(6.0 x 10^-5 C)(0.09 m²) / (3.0 x 10^-5 C)
Simplifying further, we have:
(x)² = -0.18 m²
Taking the square root of both sides, we get:
x = ±√(-0.18 m²)
Since we cannot have a negative distance, we discard the negative solution:
x = √(-0.18 m²)
The distance from the larger charge to the point where the total electric field is zero is equal to the square root of -0.18 m².