The arm \(\mathit{PQ}\) of the rectangular conductor is moved from \(x=0\) , outwards. The uniform magnetic field is perpendicular to the plane and extends from \(x=0\) to \(x=b\) and is zero for \(x>b\) . Only the arm \(\mathit{PQ}\) possesses substantial resistance \(r\) . Consider the situation when the arm \(\mathit{PQ}\) is pulled outwards from \(x=0\) to \(x=2b\) , and is then moved back to \(x=0\) with constant speed \(v\) . Obtain expressions for the flux, the induced \(\mathit{emf}\) , the force necessary to pull the arm and the power dissipated as joule heat. Sketch the variation of these quantities with distance.
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The arm \(\mathit{PQ}\) of the rectangular conductor is moved from \(x=0\) , outwards. The uniform magnetic field is perpendicular to the plane and extends from \(x=0\) to \(x=b\) and is zero for \(x>b\) . Only the arm \(\mathit{PQ}\) possesses substantial resistance \(r\) . Consider the situation when the arm \(\mathit{PQ}\) is pulled outwards from \(x=0\) to \(x=2b\) , and is then moved back to \(x=0\) with constant speed \(v\) . Obtain expressions for the flux, the induced \(\mathit{emf}\) , the force necessary to pull the arm and the power dissipated as joule heat. Sketch the variation of these quantities with distance.
Answer
Let the arm \(\mathit{PQ}\) is at distance \(^\prime x^\prime \) at any instant of time from \(x=0\) . Then
The flux linked with the circuit \(\mathit{SPQR}\)
\({\therefore}\varphi _B=\mathit{Blx}\) ` \(0{\leq}x\lt b\)
\(=\mathit{Blb}\) \(b{\leq}x<2b\)
The induced \(\mathit{emf}\) due to motion is given by:
\({\therefore}\varepsilon =\frac{-d\varphi _B}{\mathit{dt}}\)
\(=\mathit{Blv}\) \(0{\leq}x\lt b\)
\(=0\) \(b{\leq}x<2b\)
The induced current
\({\therefore}I=\frac{\varepsilon } r=\frac{\mathit{Blv}} r\)
In presence of magnetic field, the magnetic force on the arm \(\mathit{PQ}\) ,
\({\therefore}F=\mathit{BIl}\)
\(=\frac{B^2l^2v} r\) \(0{\leq}x\lt b\)
\(=0\) \(b{\leq}x<2b\)
The power dissipation
\(\therefore \;P=I^2r\)
\(=\frac{B^2l^2v^2} r\) \(0{\leq}x\lt b\)
\(=0\) \(b{\leq}x<2b\)
The flux linked with the circuit \(\mathit{SPQR}\)
\({\therefore}\varphi _B=\mathit{Blx}\) ` \(0{\leq}x\lt b\)
\(=\mathit{Blb}\) \(b{\leq}x<2b\)
The induced \(\mathit{emf}\) due to motion is given by:
\({\therefore}\varepsilon =\frac{-d\varphi _B}{\mathit{dt}}\)
\(=\mathit{Blv}\) \(0{\leq}x\lt b\)
\(=0\) \(b{\leq}x<2b\)
The induced current
\({\therefore}I=\frac{\varepsilon } r=\frac{\mathit{Blv}} r\)
In presence of magnetic field, the magnetic force on the arm \(\mathit{PQ}\) ,
\({\therefore}F=\mathit{BIl}\)
\(=\frac{B^2l^2v} r\) \(0{\leq}x\lt b\)
\(=0\) \(b{\leq}x<2b\)
The power dissipation
\(\therefore \;P=I^2r\)
\(=\frac{B^2l^2v^2} r\) \(0{\leq}x\lt b\)
\(=0\) \(b{\leq}x<2b\)
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