Parallel Currents
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Exercise:
Two straight parallel wires are placed at a distance of dO. They carry the currents IaO and IbO respectively opposite directions. Calculate the magnitude of the magnetic field at a po in the plane defined by the wires which abcliste abc has the same distance to both wires; abc has distance cm from the first and cm from the second wire. abc Find the pos in the plane where the magnetic field vanishes. abcliste
Solution:
abcliste abc The partial fields produced by the currents po in the same direction so the contributions can be added: B B_+B_ fracmu_pi leftfracI_r_+fracI_r_right labelrefB For a po in the middle between the two wires r_ r_ d/ we find B BaF fracncmuopitimes dtimes Ia+Ib Ba approx resultBaP- abc Using refrefB we find B fracncmuopitimes leftfracIara+fracIbrbright Bb approx resultBbP- abc In order for the two partial fields to cancel at some po the po is not allowed to be between the wires. It is also clear that it has to be closer to the weaker current. This means that r_r_+d. The magnitudes of B_ and B_ must be equal: B_ fracmu_pifracI_r_ fracmu_pifracI_r_ B_ This can be simplified to fracI_r_ fracI_r_ fracI_r_+d It follows that I_ r_+d I_ r_+I_ d I_ r_ Solving for r_ leads to r_ rF dOtimes fracIaIb-Ia resultrP- The combined magnetic field disappears on a line parallel to the two wires in the plane defined by the wires. It is rP- away from the weaker current in the direction away from the stronger current. abcliste
Two straight parallel wires are placed at a distance of dO. They carry the currents IaO and IbO respectively opposite directions. Calculate the magnitude of the magnetic field at a po in the plane defined by the wires which abcliste abc has the same distance to both wires; abc has distance cm from the first and cm from the second wire. abc Find the pos in the plane where the magnetic field vanishes. abcliste
Solution:
abcliste abc The partial fields produced by the currents po in the same direction so the contributions can be added: B B_+B_ fracmu_pi leftfracI_r_+fracI_r_right labelrefB For a po in the middle between the two wires r_ r_ d/ we find B BaF fracncmuopitimes dtimes Ia+Ib Ba approx resultBaP- abc Using refrefB we find B fracncmuopitimes leftfracIara+fracIbrbright Bb approx resultBbP- abc In order for the two partial fields to cancel at some po the po is not allowed to be between the wires. It is also clear that it has to be closer to the weaker current. This means that r_r_+d. The magnitudes of B_ and B_ must be equal: B_ fracmu_pifracI_r_ fracmu_pifracI_r_ B_ This can be simplified to fracI_r_ fracI_r_ fracI_r_+d It follows that I_ r_+d I_ r_+I_ d I_ r_ Solving for r_ leads to r_ rF dOtimes fracIaIb-Ia resultrP- The combined magnetic field disappears on a line parallel to the two wires in the plane defined by the wires. It is rP- away from the weaker current in the direction away from the stronger current. abcliste