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Exercise:
A short coil has a diameter of dO and NO turns. abcliste abc Show that the formula for a circular loop including the turns is a good approximation for a short coil L ll d. abc Calculate the magnetic field at the centre of the coil for a current of IO. abc How far from the centre of the coil along the central axis has the strength of the magnetic field dropped to half the maximum value? abcliste

Solution:
abcliste abc The field at the centre of a coil is given by B mu_fracN IsqrtL^+d^ For a short coil Lll d the square root can be approximated by sqrtL^+d^ dsqrtL/d^+ approx d With d r we find B &approx mu_fracN Id mu_fracN I r fracmu_ NfracIr This is the expression for the magnetic field at the centre of a circular loop with radius r and N turns. abc B BF ncmuotimesfracNtimes Id B approx resultBP- abc The field along the central axis is given by Bz fracmu_fracN I r^leftr^+z^right^frac The distance z at which the magnetic field has dropped to one half of the value at the centre of the coil is given by fracBzB frac fracr^leftr^+z^right^frac r fracr^leftr^+z^right^frac By rearranging the terms we find leftr^+z^right^frac r^ r^+z^ left r^right^frac ^frac r^ Solving for z leads to z left^frac r^-r^right^frac r left^frac-right^frac zF fracdOsqrtsqrt- resultzP- abcliste
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Exercise:
A short coil has a diameter of dO and NO turns. abcliste abc Show that the formula for a circular loop including the turns is a good approximation for a short coil L ll d. abc Calculate the magnetic field at the centre of the coil for a current of IO. abc How far from the centre of the coil along the central axis has the strength of the magnetic field dropped to half the maximum value? abcliste

Solution:
abcliste abc The field at the centre of a coil is given by B mu_fracN IsqrtL^+d^ For a short coil Lll d the square root can be approximated by sqrtL^+d^ dsqrtL/d^+ approx d With d r we find B &approx mu_fracN Id mu_fracN I r fracmu_ NfracIr This is the expression for the magnetic field at the centre of a circular loop with radius r and N turns. abc B BF ncmuotimesfracNtimes Id B approx resultBP- abc The field along the central axis is given by Bz fracmu_fracN I r^leftr^+z^right^frac The distance z at which the magnetic field has dropped to one half of the value at the centre of the coil is given by fracBzB frac fracr^leftr^+z^right^frac r fracr^leftr^+z^right^frac By rearranging the terms we find leftr^+z^right^frac r^ r^+z^ left r^right^frac ^frac r^ Solving for z leads to z left^frac r^-r^right^frac r left^frac-right^frac zF fracdOsqrtsqrt- resultzP- abcliste
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Attributes & Decorations
Branches
Magnetism
Tags
coil, current, loop, magnetic field
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Difficulty
(3, default)
Points
0 (default)
Language
ENG (English)
Type
Calculative / Quantity
Creator by
Decoration
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Link