Question

A nuclear radiation when passes through a magnetic field which is perpendicular into the paper as shown below deflects toward B. Identify the nuclear radiation.

Answer 1

Hint: When the particles of a certain nuclear radiation are brought in contact with a uniform magnetic field then it tends to undergo deflection. The Lorentz force formula is applied in order to determine which particles are associated with the deflection due to the magnetic field. The rule is also determined in order to find the direction of the velocity of the particles so that the nuclear radiation that undergoes deflection can be identified.

Complete step by step answer:
The above problem revolves around the concept of Lorentz force. In order to identify the nuclear radiation we first need to know the concept behind Lorentz forces and where it is applied. The force called the Lorentz force is applied when there is a force on moving charge particles in a magnetic field. Particles which are electrically charged experience a force due to the magnetic field applied on it and hence they are said to move with a certain velocity inside the magnetic field itself. This force which is said to deflect the charged particles sideways is known as Lorentz force.

Here, the question mentioned that a nuclear radiation is said to pass through a magnetic field and as per the question is it given that deflection occurs toward point B. The charged particles of the nuclear radiation are said to hence undergo Lorentz force which is responsible for this deflection.

The Lorentz force is given by an equation:
$\vec F = q(\vec v \times \vec B)$ ----------($1$)
Since all of these quantities, that is, force, velocity and magnetic field are all known to be vector quantities they will have magnitude as well as direction. We are required to determine the direction of each of these quantities to know why the deflection has taken place as seen in the diagram. The magnetic field considered here is said to be uniform and is given to be perpendicular, that is, in a direction is given to be going into the plane of paper.

The direction of velocity of the charged particles coming from the radiation beam are said to be perpendicular to the magnetic field in order to produce maximum force on the particles and correspondingly produce a deflection as shown in figure. The force tends to continuously deflect the charged particles sideways and hence it exhibits circular motion and hence a centripetal force is set up and we are aware that the centripetal force is said to be directed towards the center of the circular motion. Thus, this is the direction of force.

We now need to apply either the Fleming’s left hand rule or the right hand rule to determine the directions of force, field and velocity of the charged particle.Now, we apply the concept of unit vectors because these vectors are the ones that give the directions of vector quantities. Since all three are said to be mutually perpendicular we must consider a set of unit vectors in order to measure the vector quantities, that is, $\hat i$, $\hat j$ and $\hat k$ to determine the directions of force, field and velocity. Since the magnetic field is said to be produced into the plane of paper the magnetic field will be in the $ - \hat j$ direction. The velocity vector is in the $ - \hat k$ direction since it is perpendicular to the magnetic field.

Let us consider the left hand side of the equation ($1$), that is, the direction of force in terms of unit vectors. The force as discussed above is said to be in the x-axis direction, that is, along $\hat i$ direction.Hence,
$\vec F = + \hat i$ ------($3$)
Let us consider only the right hand side of the equation ($1$). By applying the concept of unit vectors into the equation ($1$) we get:
 $ \Rightarrow q( - \hat j \times - \hat k)$
$ \Rightarrow q(\hat j \times \hat k)$

The concept of cross product between these unit vectors is applied here. As per the cross product rule, the cross product of $\hat j$ and $\hat k$ is said to be $ - \hat i$. Hence the above equation becomes:
$ \Rightarrow q( - \hat i)$
Nuclear radiation is nothing but a beam of charged particles with high velocity. We know that nuclear radiation is of three types. The nuclear radiation has to be either alpha, beta or gamma radiation. Since the charged particle is said to produce deflection the radiation can be either alpha or beta radiation.

If we consider or assume the radiation to be beta radiation then the charged particles released into the uniform magnetic field will be high speed electrons since beta radiation consists of electrons only. This means that the charge of electrons considered will be negative since electrons are negatively charged particles.
Hence, the equation becomes:
$ \Rightarrow - q( - \hat i)$
By neglecting the magnitude of charge, since we are considering only the directions, we get:
$ \Rightarrow - ( - \hat i)$
$ \Rightarrow + \hat i$ ---------($4$)

Hence from equations ($3$) and ($4$) we can see that the left and right hand sides of the equation ($1$) are equal and hence LHS=RHS. This means that electrons are the particles that corresponds to the deflection in the direction of point B in order to correspond to the Lorentz force that is along $\hat i$ direction. Hence we have proved that the charged particles considered here, are the electrons and hence the nuclear radiation is identified to be beta radiation.

Note: A common error in the above problem may be the determination of the directions of the force, field and velocity of the charged particles. The correct rule for finding the direction of these quantities must be applied which is often a confusion. The left hand or the palm rule may be applied in accordance to which is more convenient and the most important point to note is that all these three quantities are mutually perpendicular to each other.