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Raen
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Charged Sphere with a Hole -- Check my work?
You have a spherical shell of radius a and charge Q. Your sphere is uniformly charged except for the region where θ<= 1° (which has σ = 0).
Imagine that your field point is somewhere on the positive z-axis (so z could be larger or smaller than a). Determine E as a function of z.
I believe I can represent this as a uniformly charged sphere without a hole and a thin disk with a charge density of -σ. Then the law of superposition let's me add the two together. I think I did it right, but before I go on to the computer program portion of the assignment, I'd love if somebody would double-check my logic and work. If you see an error, please let me know. If you think it's correct, let me know that, too.
sin(1°) = r/a Where r is the radius of the disk. r = sin(1°)a = 0.017a
E field of a sphere: E(r) = Q/(4∏r2ε0) = σa2/(r2ε0)
E field of a disk: E(z) = q/(2∏ε0(0.017a2)*(1-z2/(√z2+(0.017a)2))
Along the z-axis, the E field of the sphere can be written as E(z) = σa2/(z2ε0)
Therefore, Etotal = σa2/(z2ε0) + q/(2∏ε0(0.017a2)*(1-z2/(√z2+(0.017a)2))
or
Etotal = σ/ε0*(a2/z2 - 1/2 + z/√(z2 + (0.017a)2)
Is this correct? I'm sorry if it's messy and thank you, thank you in advance.
Homework Statement
You have a spherical shell of radius a and charge Q. Your sphere is uniformly charged except for the region where θ<= 1° (which has σ = 0).
Imagine that your field point is somewhere on the positive z-axis (so z could be larger or smaller than a). Determine E as a function of z.
I believe I can represent this as a uniformly charged sphere without a hole and a thin disk with a charge density of -σ. Then the law of superposition let's me add the two together. I think I did it right, but before I go on to the computer program portion of the assignment, I'd love if somebody would double-check my logic and work. If you see an error, please let me know. If you think it's correct, let me know that, too.
Homework Equations
sin(1°) = r/a Where r is the radius of the disk. r = sin(1°)a = 0.017a
E field of a sphere: E(r) = Q/(4∏r2ε0) = σa2/(r2ε0)
E field of a disk: E(z) = q/(2∏ε0(0.017a2)*(1-z2/(√z2+(0.017a)2))
The Attempt at a Solution
Along the z-axis, the E field of the sphere can be written as E(z) = σa2/(z2ε0)
Therefore, Etotal = σa2/(z2ε0) + q/(2∏ε0(0.017a2)*(1-z2/(√z2+(0.017a)2))
or
Etotal = σ/ε0*(a2/z2 - 1/2 + z/√(z2 + (0.017a)2)
Is this correct? I'm sorry if it's messy and thank you, thank you in advance.