How Does Gauss's Law Help Calculate Earth's Total Electric Charge?

[Lolligirl]

Homework Statement

Suppose that the electric field in the Earth's atmosphere is E = 1.50 x 10^2 N/C, pointing downward. Determine the electric charge in the Earth. (The radius of the Earth is 6371 km, and the Coulomb's constant, ke, is 8.99 x 10^9 N · m2/C2.)

Homework Equations

E=kQr/a^3
Q = E(a^3)/kr

The Attempt at a Solution

I have the equation to solve this (E = kQr/a^3) and have rearranged it to Q = E(a^3)/kr, but my problem is that I don't know what r (or a) is. The Earth's radius would be a, right? Or would it be r? If it's r, then what's a, and vice-versa? Please help me figure r and a's values out.

 

[SammyS]

Homework Statement

Suppose that the electric field in the Earth's atmosphere is E = 1.50 x 10^2 N/C, pointing downward. Determine the electric charge in the Earth. (The radius of the Earth is 6371 km, and the Coulomb's constant, ke, is 8.99 x 10^9 N · m2/C2.)

Homework Equations

E=kQr/a^3
Q = E(a^3)/kr

The Attempt at a Solution

I have the equation to solve this (E = kQr/a^3) and have rearranged it to Q = E(a^3)/kr, but my problem is that I don't know what r (or a) is. The Earth's radius would be a, right? Or would it be r? If it's r, then what's a, and vice-versa? Please help me figure r and a's values out.

Judging by the title you chose for this thread, you should be using Gauss's Law to solve this.

Can you state Gauss's Law?

 

[Lolligirl]

E=kQ/r^2. This is under the applications section of the chapter on Gauss's law, but directly after an example problem that says to find the magnitude of the electric field at a point inside the sphere, we use E=kQr/a^3. Is that different from finding the magnitude of the electric field due to a point charge in the center of the sphere?

 

[haruspex]

E=kQ/r^2. This is under the applications section of the chapter on Gauss's law, but directly after an example problem that says to find the magnitude of the electric field at a point inside the sphere, we use E=kQr/a^3. Is that different from finding the magnitude of the electric field due to a point charge in the center of the sphere?

The question says nothing about fields inside charged spheres. It mentions the field in the atmosphere (just above ground level, presumably) and charge within the sphere of the Earth.

 

[SammyS]

E=kQ/r^2. This is under the applications section of the chapter on Gauss's law, but directly after an example problem that says to find the magnitude of the electric field at a point inside the sphere, we use E=kQr/a^3. Is that different from finding the magnitude of the electric field due to a point charge in the center of the sphere?

E=kQ/r² is Coulombs Law for the electric field due to a point charge.

E=kQr/a^3 gives the electric field inside a sphere of radius, a, at a distance, r, from the sphere's center, if that sphere has total charge Q which is uniformly distributed throughout the sphere's volume. This has nothing to do with the problem stated here.

Gauss's Law can be used to show that electric field due to a spherically symmetric charge distribution is the same as the field produced by a point charge of the same value as that of the entire sphere.