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latentcorpse
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A cylindrical pipe of radius a and length L is filled with mercury of electrical conductivity [itex]\zeta[/itex]. A potential difference V acts across the two ends of the pipe, creating an electric current through the mercury (which remains stationary).
(i)Find the current density, assumed uniform, within the mercury. Using Ampere's Law, find the magnetic field [itex]\mathbf{B}[/itex] at radius r from the axis.
I said that [itex]\mathbf{J}=\zeta (\mathbf{E} + \mathbf{v} \wedge \mathbf{B})=\zeta \mathbf{E}[/itex] as [itex]\mathbf{v}=0[/itex] as mercury is stationary.
Then I took an Amperian loop around the cylinder of radius r.
Ampere's Law tells us that [itex]\oint_C \mathbf{B} \cdot \mathbf{dr}=\mu_0 \int_S \mathbf{J} \cdot \mathbf{dA}[/itex]
Then,
[itex]\oint_C \mathbf{B} \cdot \mathbf{dr} = \mu_0 |\mathbf{J}| \int_S dA \Rightarrow \mu_0 \zeta |\mathbf{E}| \pi r^2 = \oint_C \mathbf{B} \cdot \mathbf{dr}[/itex]
Firstly, how does this look up to this point?
If ok, how do I find B from that last bit?
cheers!
(i)Find the current density, assumed uniform, within the mercury. Using Ampere's Law, find the magnetic field [itex]\mathbf{B}[/itex] at radius r from the axis.
I said that [itex]\mathbf{J}=\zeta (\mathbf{E} + \mathbf{v} \wedge \mathbf{B})=\zeta \mathbf{E}[/itex] as [itex]\mathbf{v}=0[/itex] as mercury is stationary.
Then I took an Amperian loop around the cylinder of radius r.
Ampere's Law tells us that [itex]\oint_C \mathbf{B} \cdot \mathbf{dr}=\mu_0 \int_S \mathbf{J} \cdot \mathbf{dA}[/itex]
Then,
[itex]\oint_C \mathbf{B} \cdot \mathbf{dr} = \mu_0 |\mathbf{J}| \int_S dA \Rightarrow \mu_0 \zeta |\mathbf{E}| \pi r^2 = \oint_C \mathbf{B} \cdot \mathbf{dr}[/itex]
Firstly, how does this look up to this point?
If ok, how do I find B from that last bit?
cheers!