Video: Finding the Net Charge Contained in a Spherical Region from the Electric Flux through the Surface of the Region

The electric flux through a spherical surface is 4.0 × 10⁴ N⋅m²/C. What is the net charge enclosed by the surface?

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Video Transcript

The electric flux through a spherical surface is 4.0 times 10 to the fourth newtons meter squared per coulomb. What is the net charge enclosed by the surface?

We can call the flux we’re given, 4.0 times 10 to the fourth newtons meter squared per coulomb, 𝜙 sub 𝐸. This is the flux passing through a spherical surface.

And we want to solve for the net charge enclosed by that surface. We’ll call that 𝑄. If we draw a sketch of the spherical surface, we know that inside the surface there is some amount of electric charge creating electric field lines which pass out through the surface.

We’re told what the flux is through the spherical surface and want to solve for the charge that must be inside the sphere in order to create that flux. To do this, we can recall Gauss’s law.

This law tells us that the electric flux through a closed surface is equal to the charge enclosed by the surface over 𝜖 naught, the permittivity of free space, which is a constant we’ll treat as exactly 8.85 times 10 to the negative 12th farads per meter.

So if 𝜙 sub 𝐸 is equal to 𝑄 over 𝜖 naught, that means that 𝑄 is equal to 𝜙 sub 𝐸, the electric flux, times 𝜖 naught, the constant. Since we know both these values, we can plug in for them now.

When we multiply these two numbers together, we find that, to two significant figures, 𝑄 is 3.5 times 10 to the negative seventh coulombs. That’s the net electric charge enclosed by the spherical surface.

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