- #1
*Alice*
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given: the electric field at a point on the axis a distance x from the plane of a ring is [tex]E = \frac {q*x} {4*pi*E0*(x^2+r^2)^{3/2}}[/tex]
where E0
is the permeability coefficient
The charged ring is replaced by a circular sheet of charge of radius a a surface charge density sigma. The ring can be divided into infinitessimally small rings of radius r and thicknes dr. Show that the electric field is given by [tex] E= \frac {sigma} {2*E0} * [1 - \frac {x} {(x^2 + a^2)^{1/2}}][/tex]
this is what I did:
charge on each ring:
[tex] 2*pi*r*sigma*dr = A*sigma=Q [/tex]
Electric field on each ring:
[tex] E = \frac {2*pi*sigma*dr*x*r} {4*pi*E0*(x^2 + r^2)^{3/2}} = \frac {sigma*dr*x*r} {2*E0*(x^2 + r^2)^{3/2}} [/tex]
Integrate over ring:
[tex] \frac {sigma} {2*E0} * \int_{0}^{a} \frac {r} {(x^2 + r^2)^{3/2}} dx = \frac {sigma} {2*E0} * [-1/2*\frac{1} {(x^2+a^2)^{0.5}}] (from 0 to a) = \frac {sigma} {4*E0}* [1 - \frac {x} {(x^2+a^2)^{.5}}] [/tex]
why is that factor 4 here (it's supposed to be 2)? Help's very much appreciated!
LaTeX
where E0
is the permeability coefficient
The charged ring is replaced by a circular sheet of charge of radius a a surface charge density sigma. The ring can be divided into infinitessimally small rings of radius r and thicknes dr. Show that the electric field is given by [tex] E= \frac {sigma} {2*E0} * [1 - \frac {x} {(x^2 + a^2)^{1/2}}][/tex]
this is what I did:
charge on each ring:
[tex] 2*pi*r*sigma*dr = A*sigma=Q [/tex]
Electric field on each ring:
[tex] E = \frac {2*pi*sigma*dr*x*r} {4*pi*E0*(x^2 + r^2)^{3/2}} = \frac {sigma*dr*x*r} {2*E0*(x^2 + r^2)^{3/2}} [/tex]
Integrate over ring:
[tex] \frac {sigma} {2*E0} * \int_{0}^{a} \frac {r} {(x^2 + r^2)^{3/2}} dx = \frac {sigma} {2*E0} * [-1/2*\frac{1} {(x^2+a^2)^{0.5}}] (from 0 to a) = \frac {sigma} {4*E0}* [1 - \frac {x} {(x^2+a^2)^{.5}}] [/tex]
why is that factor 4 here (it's supposed to be 2)? Help's very much appreciated!
LaTeX
Last edited: