## Thursday, September 29, 2016

### Resultant Magnetic field induction

Magnetic field induction or field strength is defined as the force experienced by a unit north pole placed in a uniform magnetic field. In the first case, let us consider two identical north poles at the two corners of a equilateral triangle. We need to find the magnetic field induction at the third corner of the triangle. Each north pole applied repulsive force on the unit north pole of the third corner. This two forces are equal magnitude but they are having a certain angular separation sixty degree. To find the resultant of this two forces, we can use parallogram law of vectors and we can find the induction as shown in the diagram below.

If we place north pole at one corner and south pole of same pole strength at the other corner of the equilateral triangle, the two forces are equal in magnitude. They not only have some angular separation but also in different direction. Thus the resultant will be different as shown in the diagram below.

If identical poles are placed at the three corners of a equilateral triangle, they apply a certain force at the centroid of the triangle. This three forces are equal in magnitude but has different directions. This combination of three forces acts at the centroid and they can be redrawn as three sides of the triangle in a order. Thus they are in equilibrium and hence there is no force and induction acting at the centroid of the triangle.

If any one pole is having the same magnitude of pole strength but opposite in nature, then the resultant induction at the centroid of the triangle is not zero. The forces act with angular separation and their resultant is along the same direction of the third force. Thus by adding them, we can get the resultant force as shown in the diagram below.

We can also find the resultant magnetic induction at the cross section of two diagonals of a square as shown in the diagram below. If all poles are similar in nature and having same magnetic pole strength, the resultant induction at the cross section is zero as shown below.

If two poles at the adjacent corners are similar in nature, then the resultant induction is not zero and it can be measured as shown in the diagram below.

We can find the resultant magnetic field due to two bar magnets crossed with each other on a common line as shown in the diagram below.To the common point, one magnet will be on axial line and the other magnet will be on equatorial line. The two magnetic fields act perpendicular to each other. Thus to find their resultant, we can use parallogram law of vectors as shown in the diagram below.

If the two magnets separated by a certain distance placed parallel and opposite poles are facing each other, they generate some magnetic field around us. We would like to measure the magnetic field induction on the axial line of any one magnet. The field due to both of them is different but they are in opposite direction.  We can find the resultant as shown in the diagram below.

We can also find the magnetic field induction at a point between the two magnets connecting to their center. There are two possibilities here. Similar poles facing each other is one case and opposite poles facing each other is the other case. In the first case the effective magnetic field is the sum of the two where as in the other case, they are opposite and hence resultant field is the difference between them. It is as shown in the diagram below.

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