A spring with an unstrained length of 0.066 m and a spring constant of 2.3 N/m hangs vertically downward from the ceiling. A uniform electric field directed vertically upward fills the region containing the spring. A sphere with a mass of 4.84 10-3 kg and a net charge of +7.3 µC is attached to the lower end of the spring. The spring is released slowly, until it reaches equilibrium. The equilibrium length of the spring is 0.050 m. What is the magnitude of the external electric field?

balene the electric field force with the spring force.

dont forget gravity...

To find the magnitude of the external electric field, we can use the principle of equilibrium for the charged sphere attached to the spring. At equilibrium, the gravitational force acting downwards on the sphere is equal to the electrostatic force acting upwards on it.

First, let's calculate the gravitational force acting on the sphere:
F_gravity = m * g
where m is the mass of the sphere and g is the acceleration due to gravity.

Given:
m = 4.84 * 10^-3 kg (mass of the sphere)

g = 9.8 m/s^2 (acceleration due to gravity)

Calculating the gravitational force:
F_gravity = (4.84 * 10^-3 kg) * (9.8 m/s^2)
F_gravity = 4.7492 * 10^-2 N

Now, to find the electrostatic force, we can use Coulomb's Law:
F_electric = k * q1 * q2 / r^2
where k is the Coulomb's constant, q1 and q2 are the charges, and r is the distance between the charges.

In this case, we have only one charge (the charged sphere) and the external electric field creates a force on it. Therefore, the equation becomes:
F_electric = E_external * q
where E_external is the magnitude of the external electric field and q is the charge.

Given:
q = +7.3 µC (charge of the sphere)

We need to convert the charge into coulombs:
1 µC = 10^-6 C
Therefore,
q = +7.3 * 10^-6 C

Now, we can set up the equilibrium equation:
F_gravity = F_electric

Substituting the values, we have:
4.7492 * 10^-2 N = E_external * (7.3 * 10^-6 C)

Now, we can solve for E_external:
E_external = (4.7492 * 10^-2 N) / (7.3 * 10^-6 C)

Calculating:
E_external = 6.5115 * 10^3 N/C

Therefore, the magnitude of the external electric field is approximately 6.5115 * 10^3 N/C.