malindenmoyer
- 30
- 0
Two very large parallel conducting plates of very large length [tex]l[/tex], and width [tex]w[/tex] are separated by a distance [tex]d[/tex]. A current [tex]I=Jw[/tex] flows to the right in the lower plate and to the left in the upper plate. Each of the two currents produces a magnetic field [tex]\frac{B}{2}[/tex] between the two plates.
(a) Show that the total field between the plates is [tex]B=\mu_0 J[/tex] via Ampere's Circuital Law.
(b) Find the flux [tex]\phi[/tex] and the self inductance per unit lenght, [tex]\frac{L}{l}[/tex] for this arrangement.
(c) Find the capacitance per unit length, [tex]\frac{C}{l}[/tex].
(d) Find [tex]\sqrt{\frac{L}{C}}[/tex] in terms of [tex]\mu_0, \epsilon_0[/tex] and geometrical factors
My Attempt at Solution
Part (a) is confusing as I have not used the circuital Law for rectangular geometry. I know that Ampere's Law is given by:
[tex]\oint_C \mathbf{B} \cdot \mathrm{d}\boldsymbol{\ell} = \mu_0 I_{\mathrm{enc}}[/tex]
But am confused as to how to apply it as it is not circular geometry.
In part (b) I know that:
[tex]\phi=BA[/tex]
But am not sure as to what area to use, since we know B per part (a)
Solving for flux leads us one step closer to finding the self inductance which is:
[tex]L=\frac{N\phi}{I}[/tex]
But again, I do not know what value to substitute in for [tex]N[/tex].
I am pretty sure I can find the capacitance per unit length per (c), and then (d) is a matter of combing (b) and (c) so that would be self explanatory. Could somebody help me get this thing started? Please keep in mind that I have a very elementary understanding of this material.
Thanks.
(a) Show that the total field between the plates is [tex]B=\mu_0 J[/tex] via Ampere's Circuital Law.
(b) Find the flux [tex]\phi[/tex] and the self inductance per unit lenght, [tex]\frac{L}{l}[/tex] for this arrangement.
(c) Find the capacitance per unit length, [tex]\frac{C}{l}[/tex].
(d) Find [tex]\sqrt{\frac{L}{C}}[/tex] in terms of [tex]\mu_0, \epsilon_0[/tex] and geometrical factors
My Attempt at Solution
Part (a) is confusing as I have not used the circuital Law for rectangular geometry. I know that Ampere's Law is given by:
[tex]\oint_C \mathbf{B} \cdot \mathrm{d}\boldsymbol{\ell} = \mu_0 I_{\mathrm{enc}}[/tex]
But am confused as to how to apply it as it is not circular geometry.
In part (b) I know that:
[tex]\phi=BA[/tex]
But am not sure as to what area to use, since we know B per part (a)
Solving for flux leads us one step closer to finding the self inductance which is:
[tex]L=\frac{N\phi}{I}[/tex]
But again, I do not know what value to substitute in for [tex]N[/tex].
I am pretty sure I can find the capacitance per unit length per (c), and then (d) is a matter of combing (b) and (c) so that would be self explanatory. Could somebody help me get this thing started? Please keep in mind that I have a very elementary understanding of this material.
Thanks.