Helmholtz coil force
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That being said... How many "default points" should you associate with an exercise upon creation?
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When you click an exercise into a collection, this number will be taken as difficulty for the exercise, kind of "by default".
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That being said... How "difficult" is an exercise? It depends on many factors, like what was being taught etc.
In physics exercises, we try to follow this pattern:
Level 1 - One formula (one you would find in a reference book) is enough to solve the exercise. Example exercise
Level 2 - Two formulas are needed, it's possible to compute an "in-between" solution, i.e. no algebraic equation needed. Example exercise
Level 3 - "Chain-computations" like on level 2, but 3+ calculations. Still, no equations, i.e. you are not forced to solve it in an algebraic manner. Example exercise
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Exercise:
Discuss how the direction and magnitude of the force between the two coils of a Helmholtz setup can be found.
Solution:
A Helmholtz setup consists of two identical coaxial coils radius R N turns each at a distance dR carrying the same current I in the same direction. The force on each coil is the of the Lorentz forces on all its wire elements caused by the magnetic field of the other coil. medskip textbfDirection of the force. We look at the force on coil which is exposed to the field vec B_ of coil . Near coil the field lines of coil are not parallel to the axis since they spread out on their way from the inside of the coil to the outside. The field at the location of a wire of coil therefore has two components: itemize item an axial component B_x parallel to the axis item a radial component B_r poing away from the axis. itemize The force on a wire element with length mathrmdvec l is mathrmdvec F Imathrmdvec ltimesvec B_. The current flows in circles around the axis so mathrmdvec l is perpicular to both components: itemize item mathrmdvec ltimesvec B_x pos radially. These forces try to stretch the coil but they po in opposite directions on opposite sides of the coil and cancel in the total force. item mathrmdvec ltimesvec B_r pos along the axis. These forces have the same direction all around the coil and add up. itemize The sketch below shows a cross-section through the axis. With the current poing out of the plane at the top wires and o the plane at the bottom wires the right-hand rule gives vec B_ poing to the right on the axis. At the top wire of coil the radial component pos upwards and hat ztimeshat y-hat x so the force pos towards coil . At the bottom wire both the current and B_r are reversed and the force pos to the left again. center tikzpicturestealth scale. % axis drawdashed gray - -- . noderight axis; % field lines of coil drawblue - -- .; drawblue - . .. controls . and . .. ..; drawblue - -. .. controls -. and -. .. .-.; % coil wires cross-section draw . circle .; fill . circle .; draw -. circle .; draw -.-. -- .-.; draw -.-. -- .-.; nodeabove left at . coil ; % coil wires cross-section draw .. circle .; fill .. circle .; draw .-. circle .; draw .-. -- .-.; draw .-. -- .-.; nodeabove right at .. coil ; % field at top wire of coil and its components drawblue - thick .. -- .. noderight vec B_; drawblue - dashed .. -- .. noderight B_x; drawblue - dashed .. -- .. nodeabove B_r; drawblue - dashed .-. -- .-. nodebelow B_r; % forces drawred - ultra thick .. -- .. nodemidway below vec F; drawred - ultra thick .-. -- .-. nodemidway above vec F; % distance draw- -. -- .-. nodemidway below dR; node at -.. Iodot; node at -.-. Iotimes; tikzpicture center The force on coil therefore pos towards coil : the coils attract each other. According to Newton's third law coil is pulled towards coil by a force of the same magnitude. This agrees with the rule for two parallel wires: parallel currents attract each other. If the current in one coil is reversed the force is reversed and the coils repel each other. medskip textbfMagnitude of the force. Only the axial component of the force adds up. The length of the wire is pi R per turn and B_r is the same everywhere along the circle by symmetry so F N I pi R B_r where B_r is the radial component of the field of coil at the location of the wire of coil . It is proportional to the current and the number of turns of coil so Fpropto N^ I^.
Discuss how the direction and magnitude of the force between the two coils of a Helmholtz setup can be found.
Solution:
A Helmholtz setup consists of two identical coaxial coils radius R N turns each at a distance dR carrying the same current I in the same direction. The force on each coil is the of the Lorentz forces on all its wire elements caused by the magnetic field of the other coil. medskip textbfDirection of the force. We look at the force on coil which is exposed to the field vec B_ of coil . Near coil the field lines of coil are not parallel to the axis since they spread out on their way from the inside of the coil to the outside. The field at the location of a wire of coil therefore has two components: itemize item an axial component B_x parallel to the axis item a radial component B_r poing away from the axis. itemize The force on a wire element with length mathrmdvec l is mathrmdvec F Imathrmdvec ltimesvec B_. The current flows in circles around the axis so mathrmdvec l is perpicular to both components: itemize item mathrmdvec ltimesvec B_x pos radially. These forces try to stretch the coil but they po in opposite directions on opposite sides of the coil and cancel in the total force. item mathrmdvec ltimesvec B_r pos along the axis. These forces have the same direction all around the coil and add up. itemize The sketch below shows a cross-section through the axis. With the current poing out of the plane at the top wires and o the plane at the bottom wires the right-hand rule gives vec B_ poing to the right on the axis. At the top wire of coil the radial component pos upwards and hat ztimeshat y-hat x so the force pos towards coil . At the bottom wire both the current and B_r are reversed and the force pos to the left again. center tikzpicturestealth scale. % axis drawdashed gray - -- . noderight axis; % field lines of coil drawblue - -- .; drawblue - . .. controls . and . .. ..; drawblue - -. .. controls -. and -. .. .-.; % coil wires cross-section draw . circle .; fill . circle .; draw -. circle .; draw -.-. -- .-.; draw -.-. -- .-.; nodeabove left at . coil ; % coil wires cross-section draw .. circle .; fill .. circle .; draw .-. circle .; draw .-. -- .-.; draw .-. -- .-.; nodeabove right at .. coil ; % field at top wire of coil and its components drawblue - thick .. -- .. noderight vec B_; drawblue - dashed .. -- .. noderight B_x; drawblue - dashed .. -- .. nodeabove B_r; drawblue - dashed .-. -- .-. nodebelow B_r; % forces drawred - ultra thick .. -- .. nodemidway below vec F; drawred - ultra thick .-. -- .-. nodemidway above vec F; % distance draw- -. -- .-. nodemidway below dR; node at -.. Iodot; node at -.-. Iotimes; tikzpicture center The force on coil therefore pos towards coil : the coils attract each other. According to Newton's third law coil is pulled towards coil by a force of the same magnitude. This agrees with the rule for two parallel wires: parallel currents attract each other. If the current in one coil is reversed the force is reversed and the coils repel each other. medskip textbfMagnitude of the force. Only the axial component of the force adds up. The length of the wire is pi R per turn and B_r is the same everywhere along the circle by symmetry so F N I pi R B_r where B_r is the radial component of the field of coil at the location of the wire of coil . It is proportional to the current and the number of turns of coil so Fpropto N^ I^.
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Exercise:
Discuss how the direction and magnitude of the force between the two coils of a Helmholtz setup can be found.
Solution:
A Helmholtz setup consists of two identical coaxial coils radius R N turns each at a distance dR carrying the same current I in the same direction. The force on each coil is the of the Lorentz forces on all its wire elements caused by the magnetic field of the other coil. medskip textbfDirection of the force. We look at the force on coil which is exposed to the field vec B_ of coil . Near coil the field lines of coil are not parallel to the axis since they spread out on their way from the inside of the coil to the outside. The field at the location of a wire of coil therefore has two components: itemize item an axial component B_x parallel to the axis item a radial component B_r poing away from the axis. itemize The force on a wire element with length mathrmdvec l is mathrmdvec F Imathrmdvec ltimesvec B_. The current flows in circles around the axis so mathrmdvec l is perpicular to both components: itemize item mathrmdvec ltimesvec B_x pos radially. These forces try to stretch the coil but they po in opposite directions on opposite sides of the coil and cancel in the total force. item mathrmdvec ltimesvec B_r pos along the axis. These forces have the same direction all around the coil and add up. itemize The sketch below shows a cross-section through the axis. With the current poing out of the plane at the top wires and o the plane at the bottom wires the right-hand rule gives vec B_ poing to the right on the axis. At the top wire of coil the radial component pos upwards and hat ztimeshat y-hat x so the force pos towards coil . At the bottom wire both the current and B_r are reversed and the force pos to the left again. center tikzpicturestealth scale. % axis drawdashed gray - -- . noderight axis; % field lines of coil drawblue - -- .; drawblue - . .. controls . and . .. ..; drawblue - -. .. controls -. and -. .. .-.; % coil wires cross-section draw . circle .; fill . circle .; draw -. circle .; draw -.-. -- .-.; draw -.-. -- .-.; nodeabove left at . coil ; % coil wires cross-section draw .. circle .; fill .. circle .; draw .-. circle .; draw .-. -- .-.; draw .-. -- .-.; nodeabove right at .. coil ; % field at top wire of coil and its components drawblue - thick .. -- .. noderight vec B_; drawblue - dashed .. -- .. noderight B_x; drawblue - dashed .. -- .. nodeabove B_r; drawblue - dashed .-. -- .-. nodebelow B_r; % forces drawred - ultra thick .. -- .. nodemidway below vec F; drawred - ultra thick .-. -- .-. nodemidway above vec F; % distance draw- -. -- .-. nodemidway below dR; node at -.. Iodot; node at -.-. Iotimes; tikzpicture center The force on coil therefore pos towards coil : the coils attract each other. According to Newton's third law coil is pulled towards coil by a force of the same magnitude. This agrees with the rule for two parallel wires: parallel currents attract each other. If the current in one coil is reversed the force is reversed and the coils repel each other. medskip textbfMagnitude of the force. Only the axial component of the force adds up. The length of the wire is pi R per turn and B_r is the same everywhere along the circle by symmetry so F N I pi R B_r where B_r is the radial component of the field of coil at the location of the wire of coil . It is proportional to the current and the number of turns of coil so Fpropto N^ I^.
Discuss how the direction and magnitude of the force between the two coils of a Helmholtz setup can be found.
Solution:
A Helmholtz setup consists of two identical coaxial coils radius R N turns each at a distance dR carrying the same current I in the same direction. The force on each coil is the of the Lorentz forces on all its wire elements caused by the magnetic field of the other coil. medskip textbfDirection of the force. We look at the force on coil which is exposed to the field vec B_ of coil . Near coil the field lines of coil are not parallel to the axis since they spread out on their way from the inside of the coil to the outside. The field at the location of a wire of coil therefore has two components: itemize item an axial component B_x parallel to the axis item a radial component B_r poing away from the axis. itemize The force on a wire element with length mathrmdvec l is mathrmdvec F Imathrmdvec ltimesvec B_. The current flows in circles around the axis so mathrmdvec l is perpicular to both components: itemize item mathrmdvec ltimesvec B_x pos radially. These forces try to stretch the coil but they po in opposite directions on opposite sides of the coil and cancel in the total force. item mathrmdvec ltimesvec B_r pos along the axis. These forces have the same direction all around the coil and add up. itemize The sketch below shows a cross-section through the axis. With the current poing out of the plane at the top wires and o the plane at the bottom wires the right-hand rule gives vec B_ poing to the right on the axis. At the top wire of coil the radial component pos upwards and hat ztimeshat y-hat x so the force pos towards coil . At the bottom wire both the current and B_r are reversed and the force pos to the left again. center tikzpicturestealth scale. % axis drawdashed gray - -- . noderight axis; % field lines of coil drawblue - -- .; drawblue - . .. controls . and . .. ..; drawblue - -. .. controls -. and -. .. .-.; % coil wires cross-section draw . circle .; fill . circle .; draw -. circle .; draw -.-. -- .-.; draw -.-. -- .-.; nodeabove left at . coil ; % coil wires cross-section draw .. circle .; fill .. circle .; draw .-. circle .; draw .-. -- .-.; draw .-. -- .-.; nodeabove right at .. coil ; % field at top wire of coil and its components drawblue - thick .. -- .. noderight vec B_; drawblue - dashed .. -- .. noderight B_x; drawblue - dashed .. -- .. nodeabove B_r; drawblue - dashed .-. -- .-. nodebelow B_r; % forces drawred - ultra thick .. -- .. nodemidway below vec F; drawred - ultra thick .-. -- .-. nodemidway above vec F; % distance draw- -. -- .-. nodemidway below dR; node at -.. Iodot; node at -.-. Iotimes; tikzpicture center The force on coil therefore pos towards coil : the coils attract each other. According to Newton's third law coil is pulled towards coil by a force of the same magnitude. This agrees with the rule for two parallel wires: parallel currents attract each other. If the current in one coil is reversed the force is reversed and the coils repel each other. medskip textbfMagnitude of the force. Only the axial component of the force adds up. The length of the wire is pi R per turn and B_r is the same everywhere along the circle by symmetry so F N I pi R B_r where B_r is the radial component of the field of coil at the location of the wire of coil . It is proportional to the current and the number of turns of coil so Fpropto N^ I^.
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Lecture Notes Magnetism by by
| Title | Matched on |
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| Helmholtz coil | tagstitle |
| Measuring Earth's magnetic field | tags |