8.1 Magnetic Force
Earlier we have seen that the source of magnetic field is a single moving charge, or bunch of moving charges, in other words, electric current. We have seen that a moving charge or a current element generates the magnetic field. And, if we place any other moving charge, or current carrying wire into this magnetic field, now we will see that this magnetic field will exert a force on that moving charge or moving charges.
We first studied the first part of these interactions, and we have seen that how the magnetic field is generated from it’s source through Biot-Savart law. And, it was such that magnetic field is equal to μ0 over 4π integral of i dl cross r over r3, for a current carrying wire. Or, for a single charge, it was in the form of μ0 over 4π qv cross unit vector r̂ divided by r2.
So, these were the two different forms of Biot-Savart law which happens to calculate the magnetic field from it’s source. Also we have introduced Ampere’s law, which was given as integral of B dot dl integrated over a closed loop, or a closed contour, which was equal to μ0 times the net current flowing through the area surrounded by this loop.
Now, by applying these two laws, we were able to calculate the magnetic field from it’s source. Now, we’re going to look at the second part of these interactions of how this external magnetic field exerts a force on any other moving charge or a current carrying conductor placed into that region.
The force, the magnetic force exerted by an external magnetic field on a single current carrying charge is given in the form of FB is equal to qv cross B. That’s the mathematical expression for the magnetic force acting on a moving charge in an external magnetic field. So, key point over here is this magnetic field is an external magnetic field.
In other words, it is not the magnetic field generated by this moving charge. Again, as you recall, the electric field of a charge, will not exert a force on that charge. In a similar way, magnetic field of a moving charge will not exert a force on that charge. It will exert a force on other moving charge or charges.
Alright, if we first consider this force, the magnetic force, FB, since it is equal to qv cross B, we see that the magnitude of this force is going to be equal to magnitude of the first vector, and that is qv times the magnitude of the second vector B and times the sine of the angle between these two vectors.
Now, if you consider a positive charge, for example, moving with a velocity of v placed in a magnetic field region such that, let’s say, the B points in this direction, then the angle between qv and B is this angle. And, therefore the magnitude of this force is the magnitude of qv times the magnitude of the magnetic field vector times sine of the angle between these two vectors.
The direction of this force will be determined from the right hand rule, because it is equal to the cross product of these two vectors. And, holding, in this case, right hand fingers in the direction of first vector qv, pointing in this direction, and adjusting them to curl towards this second vector, B, while keeping the thumb in up position, or in open position, we will see that for this case, the force on this charge exerted by this magnetic field will be pointing into the plane. In other words, the force on this charge is going to be pointing, for this configuration, into the plane.
Or, if you consider another example, such that, let’s say qv is in this direction and the magnetic field, B, is in this direction, and let’s assume that they represent a plane which is perpendicular to the plane of the the screen, in a three dimensional sense, something like this. Then, if we look at the qv cross B force, by applying the right hand rule, is going to be acting on this charge in this direction. In other words, perpendicular to the plane represented by v and B. So, this force is going to be perpendicular to the direction of both the magnetic field and as well as the velocity vector.
Of course, this is originating directly from the properties of the cross products. Well, so if we, then, write down the properties of the magnetic force, first we can say that the magnitude of FB is proportional to charge, the magnitude of the charge, and the magnitude of the velocity vector. And, it is proportional to the strength of the magnetic field. And, therefore, any time whenever one of these quantities, or all of them increases, then the force generated by the magnetic field on that moving charge will also increase. So, FB increases as qv and B increases.
And, second property, we can say that FB is perpendicular to vector qv, the velocity vector, and as well as it is perpendicular to the magnetic field vector. So, the force vector generated by the magnetic field is going to be always perpendicular to the plane represented by the velocity vector and the magnetic field vector. Moving on, if we look at the magnitude expression over here, the force will take it’s maximum value whenever the right hand side takes it’s maximum value, and that occurs when sine of