Mixing
I need help solving this problem regarding charge conservation and electric current.
1
$begingroup$
Problem
I think that part (a) is asking "What can be inferred by applying conservation of charge to an arbitrary control volume?
I found the following breakdown that I think helps answer part (a), but I have no idea how to do part (b).
Solution to part (a)
Thanks guys!
pde physics mathematical-physics
edited Feb 2 at 19:23
Robert Lewis
49k23168
asked Feb 2 at 19:21
JasenJasen
61
$endgroup$
add a comment |
1
$begingroup$
Problem
I think that part (a) is asking "What can be inferred by applying conservation of charge to an arbitrary control volume?
I found the following breakdown that I think helps answer part (a), but I have no idea how to do part (b).
Solution to part (a)
Thanks guys!
pde physics mathematical-physics
edited Feb 2 at 19:23
Robert Lewis
49k23168
asked Feb 2 at 19:21
JasenJasen
61
$endgroup$
$begingroup$
I added the "physics" and "mathematical-physics" tags to your post. Cheers!
$endgroup$
– Robert Lewis
Feb 2 at 19:24
add a comment |
1
1
1
$begingroup$
Problem
I think that part (a) is asking "What can be inferred by applying conservation of charge to an arbitrary control volume?
I found the following breakdown that I think helps answer part (a), but I have no idea how to do part (b).
Solution to part (a)
Thanks guys!
pde physics mathematical-physics
edited Feb 2 at 19:23
Robert Lewis
49k23168
asked Feb 2 at 19:21
JasenJasen
61
$endgroup$
Problem
I think that part (a) is asking "What can be inferred by applying conservation of charge to an arbitrary control volume?
I found the following breakdown that I think helps answer part (a), but I have no idea how to do part (b).
Solution to part (a)
Thanks guys!
pde physics mathematical-physics
pde physics mathematical-physics
edited Feb 2 at 19:23
Robert Lewis
49k23168
asked Feb 2 at 19:21
JasenJasen
61
edited Feb 2 at 19:23
Robert Lewis
49k23168
asked Feb 2 at 19:21
JasenJasen
61
edited Feb 2 at 19:23
Robert Lewis
49k23168
edited Feb 2 at 19:23
Robert Lewis
49k23168
edited Feb 2 at 19:23
Robert Lewis
49k23168
49k23168
asked Feb 2 at 19:21
JasenJasen
61
asked Feb 2 at 19:21
JasenJasen
61
asked Feb 2 at 19:21
JasenJasen
61
61
$begingroup$
I added the "physics" and "mathematical-physics" tags to your post. Cheers!
$endgroup$
– Robert Lewis
Feb 2 at 19:24
add a comment |
$begingroup$
I added the "physics" and "mathematical-physics" tags to your post. Cheers!
$endgroup$
– Robert Lewis
Feb 2 at 19:24
$begingroup$
I added the "physics" and "mathematical-physics" tags to your post. Cheers!
$endgroup$
– Robert Lewis
Feb 2 at 19:24
$begingroup$
I added the "physics" and "mathematical-physics" tags to your post. Cheers!
$endgroup$
– Robert Lewis
Feb 2 at 19:24
add a comment |
1 Answer
1
active
oldest
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By electroneutrality, $0approxpartial_trho_e=-vec{nabla}cdotvec{J}=kappa_enabla^2phi$.
answered Feb 2 at 19:48
J.G.J.G.
33.5k23252
$endgroup$
$begingroup$
Can you explain what you mean by that? And does that answer part b?
$endgroup$
– Jasen
Feb 2 at 19:51
$begingroup$
@Jasen It is (b), yes. Electroneutrality is only needed to approximate any of this as $0$.
$endgroup$
– J.G.
Feb 2 at 20:04
$begingroup$
Thank you. I'm afraid I still don't totally understand how this shows the PDE that governs Φ(r,t) though. I understand that electroneutrality causes ρ to be 0. But where is the ∂t coming from? Also, how exactly are the two equations getting tied together?
$endgroup$
– Jasen
Feb 2 at 20:11
$begingroup$
@Jasen Technically it's possible for $rho_e$ to be small but $partial_trho_e$ not to be, if e.g. $rho_e$ oscillates at high frequency, but a physicist expects this not to apply in conditions described as electroneutral. The rest is the continuity equation.
$endgroup$
– J.G.
Feb 2 at 20:33
add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
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active
oldest
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0
$begingroup$
By electroneutrality, $0approxpartial_trho_e=-vec{nabla}cdotvec{J}=kappa_enabla^2phi$.
answered Feb 2 at 19:48
J.G.J.G.
33.5k23252
$endgroup$
$begingroup$
Can you explain what you mean by that? And does that answer part b?
$endgroup$
– Jasen
Feb 2 at 19:51
$begingroup$
@Jasen It is (b), yes. Electroneutrality is only needed to approximate any of this as $0$.
$endgroup$
– J.G.
Feb 2 at 20:04
$begingroup$
Thank you. I'm afraid I still don't totally understand how this shows the PDE that governs Φ(r,t) though. I understand that electroneutrality causes ρ to be 0. But where is the ∂t coming from? Also, how exactly are the two equations getting tied together?
$endgroup$
– Jasen
Feb 2 at 20:11
$begingroup$
@Jasen Technically it's possible for $rho_e$ to be small but $partial_trho_e$ not to be, if e.g. $rho_e$ oscillates at high frequency, but a physicist expects this not to apply in conditions described as electroneutral. The rest is the continuity equation.
$endgroup$
– J.G.
Feb 2 at 20:33
add a comment |
0
$begingroup$
By electroneutrality, $0approxpartial_trho_e=-vec{nabla}cdotvec{J}=kappa_enabla^2phi$.
answered Feb 2 at 19:48
J.G.J.G.
33.5k23252
$endgroup$
$begingroup$
Can you explain what you mean by that? And does that answer part b?
$endgroup$
– Jasen
Feb 2 at 19:51
$begingroup$
@Jasen It is (b), yes. Electroneutrality is only needed to approximate any of this as $0$.
$endgroup$
– J.G.
Feb 2 at 20:04
$begingroup$
Thank you. I'm afraid I still don't totally understand how this shows the PDE that governs Φ(r,t) though. I understand that electroneutrality causes ρ to be 0. But where is the ∂t coming from? Also, how exactly are the two equations getting tied together?
$endgroup$
– Jasen
Feb 2 at 20:11
$begingroup$
@Jasen Technically it's possible for $rho_e$ to be small but $partial_trho_e$ not to be, if e.g. $rho_e$ oscillates at high frequency, but a physicist expects this not to apply in conditions described as electroneutral. The rest is the continuity equation.
$endgroup$
– J.G.
Feb 2 at 20:33
add a comment |
0
0
0
$begingroup$
By electroneutrality, $0approxpartial_trho_e=-vec{nabla}cdotvec{J}=kappa_enabla^2phi$.
answered Feb 2 at 19:48
J.G.J.G.
33.5k23252
$endgroup$
By electroneutrality, $0approxpartial_trho_e=-vec{nabla}cdotvec{J}=kappa_enabla^2phi$.
answered Feb 2 at 19:48
J.G.J.G.
33.5k23252
answered Feb 2 at 19:48
J.G.J.G.
33.5k23252
answered Feb 2 at 19:48
J.G.J.G.
33.5k23252
answered Feb 2 at 19:48
J.G.J.G.
33.5k23252
33.5k23252
$begingroup$
Can you explain what you mean by that? And does that answer part b?
$endgroup$
– Jasen
Feb 2 at 19:51
$begingroup$
@Jasen It is (b), yes. Electroneutrality is only needed to approximate any of this as $0$.
$endgroup$
– J.G.
Feb 2 at 20:04
$begingroup$
Thank you. I'm afraid I still don't totally understand how this shows the PDE that governs Φ(r,t) though. I understand that electroneutrality causes ρ to be 0. But where is the ∂t coming from? Also, how exactly are the two equations getting tied together?
$endgroup$
– Jasen
Feb 2 at 20:11
$begingroup$
@Jasen Technically it's possible for $rho_e$ to be small but $partial_trho_e$ not to be, if e.g. $rho_e$ oscillates at high frequency, but a physicist expects this not to apply in conditions described as electroneutral. The rest is the continuity equation.
$endgroup$
– J.G.
Feb 2 at 20:33
add a comment |
$begingroup$
Can you explain what you mean by that? And does that answer part b?
$endgroup$
– Jasen
Feb 2 at 19:51
$begingroup$
@Jasen It is (b), yes. Electroneutrality is only needed to approximate any of this as $0$.
$endgroup$
– J.G.
Feb 2 at 20:04
$begingroup$
Thank you. I'm afraid I still don't totally understand how this shows the PDE that governs Φ(r,t) though. I understand that electroneutrality causes ρ to be 0. But where is the ∂t coming from? Also, how exactly are the two equations getting tied together?
$endgroup$
– Jasen
Feb 2 at 20:11
$begingroup$
@Jasen Technically it's possible for $rho_e$ to be small but $partial_trho_e$ not to be, if e.g. $rho_e$ oscillates at high frequency, but a physicist expects this not to apply in conditions described as electroneutral. The rest is the continuity equation.
$endgroup$
– J.G.
Feb 2 at 20:33
$begingroup$
Can you explain what you mean by that? And does that answer part b?
$endgroup$
– Jasen
Feb 2 at 19:51
$begingroup$
Can you explain what you mean by that? And does that answer part b?
$endgroup$
– Jasen
Feb 2 at 19:51
$begingroup$
@Jasen It is (b), yes. Electroneutrality is only needed to approximate any of this as $0$.
$endgroup$
– J.G.
Feb 2 at 20:04
$begingroup$
@Jasen It is (b), yes. Electroneutrality is only needed to approximate any of this as $0$.
$endgroup$
– J.G.
Feb 2 at 20:04
$begingroup$
Thank you. I'm afraid I still don't totally understand how this shows the PDE that governs Φ(r,t) though. I understand that electroneutrality causes ρ to be 0. But where is the ∂t coming from? Also, how exactly are the two equations getting tied together?
$endgroup$
– Jasen
Feb 2 at 20:11
$begingroup$
Thank you. I'm afraid I still don't totally understand how this shows the PDE that governs Φ(r,t) though. I understand that electroneutrality causes ρ to be 0. But where is the ∂t coming from? Also, how exactly are the two equations getting tied together?
$endgroup$
– Jasen
Feb 2 at 20:11
$begingroup$
@Jasen Technically it's possible for $rho_e$ to be small but $partial_trho_e$ not to be, if e.g. $rho_e$ oscillates at high frequency, but a physicist expects this not to apply in conditions described as electroneutral. The rest is the continuity equation.
$endgroup$
– J.G.
Feb 2 at 20:33
$begingroup$
@Jasen Technically it's possible for $rho_e$ to be small but $partial_trho_e$ not to be, if e.g. $rho_e$ oscillates at high frequency, but a physicist expects this not to apply in conditions described as electroneutral. The rest is the continuity equation.
$endgroup$
– J.G.
Feb 2 at 20:33
add a comment |
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$begingroup$
I added the "physics" and "mathematical-physics" tags to your post. Cheers!
$endgroup$
– Robert Lewis
Feb 2 at 19:24