PG1995
Active Member
Hi
I understand it's a long post and perhaps I should have made the queries in separate posts. But all three queries are related so I thought it would be a better idea to combine them in one post. It would be really kind of you if you could help me. Thank you for your time and help.
Magnetic field of an infinite straight current carrying conductor is given as: [latex]B=\frac{\mu _{0}I}{2\pi r}[/latex].
Likewise, electric field of an infinite line of charge is also given by a similar formula: [latex]E=\frac{\lambda }{2\pi \epsilon _{0}R}[/latex], "λ" is linear charge density, λ=Q/2a.
Magnetic field at the center of N circular loops is given as: [latex]B=\frac{\mu _{0}NI}{2a}[/latex], where "N" is number of loops and "a" is radius.
Q: Suppose we have a loop with an infinitesimal radius dr. It would be reasonable to assume that field over the cross section of the loop is constant, at least for the radius dr. The circumference of the loop is: 2π(dr). Assume that this circumference can be subdivided into infinitesimally small 10 segments, dl's, i.e. 10(dl)=2π(dr). When radius is double, the circumference gets doubled, i.e. circumference for 2dr radius=2π(2dr)=4π(dr). This means that dl segments carved out of the circumference also get doubled, i.e. 2(10)=20. This might lead one to erroneously conclude that magnetic field in the cross section of the loop is still constant because even though radius has been doubled but there have also been twice more segments to contribute to the field. But here one should consider the area of a circle which does have linear relationship with radius: area=πr^2. It means for each increase in radius area would increase more as compared to a simple linear relation. But still don't you think that the magnetic field at the very center of the loop should remain constant?
You can find Q1 and Q2 included in **broken link removed** attachment. By the way, in Q2, I mention a picture at the very beginning, you can find it **broken link removed**.
Regards
PG
I understand it's a long post and perhaps I should have made the queries in separate posts. But all three queries are related so I thought it would be a better idea to combine them in one post. It would be really kind of you if you could help me. Thank you for your time and help.
Magnetic field of an infinite straight current carrying conductor is given as: [latex]B=\frac{\mu _{0}I}{2\pi r}[/latex].
Likewise, electric field of an infinite line of charge is also given by a similar formula: [latex]E=\frac{\lambda }{2\pi \epsilon _{0}R}[/latex], "λ" is linear charge density, λ=Q/2a.
Magnetic field at the center of N circular loops is given as: [latex]B=\frac{\mu _{0}NI}{2a}[/latex], where "N" is number of loops and "a" is radius.
Q: Suppose we have a loop with an infinitesimal radius dr. It would be reasonable to assume that field over the cross section of the loop is constant, at least for the radius dr. The circumference of the loop is: 2π(dr). Assume that this circumference can be subdivided into infinitesimally small 10 segments, dl's, i.e. 10(dl)=2π(dr). When radius is double, the circumference gets doubled, i.e. circumference for 2dr radius=2π(2dr)=4π(dr). This means that dl segments carved out of the circumference also get doubled, i.e. 2(10)=20. This might lead one to erroneously conclude that magnetic field in the cross section of the loop is still constant because even though radius has been doubled but there have also been twice more segments to contribute to the field. But here one should consider the area of a circle which does have linear relationship with radius: area=πr^2. It means for each increase in radius area would increase more as compared to a simple linear relation. But still don't you think that the magnetic field at the very center of the loop should remain constant?
You can find Q1 and Q2 included in **broken link removed** attachment. By the way, in Q2, I mention a picture at the very beginning, you can find it **broken link removed**.
Regards
PG
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