### BdPhO Regional (Dhaka-South) : Problemsets

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**Fri Apr 26, 2019 5:25 pm**$***BdPhO$ $Regional$ $(Dhaka-South)$ $: Higher$ $Secondary***$

A coherent beam of monochromatic light of wavelength $500$ $nm$ is directed towards two thin slits separated by a distance of $1$ $mm$. If the screen covers an angular region of $60˚$, as shown in the figure, approximately how many bright interference bands are visible on the screen?

Consider a conducting sphere with charge $+q$ and a point charge $-q$ kept as shown in the figure. Draw electric field lines.

A large loop of radius $R$ carries a current of $I$. A much smaller loop of radius $r$ is placed at the center of this loop, such that its axis is perpendicular to that of the larger loop. The smaller loop carries a much smaller current $i$. What is the torque on the smaller loop?

Assume we have a long column of water and an air bubble of radius $r$ and pressure $P_0$ at the surface of the Earth is rising inside that. If the column is long enough for $g$ not to be considered constant, find $r$ as a function of $R$; where $R$ is the distance between the bubble and the center of Earth. [The radius of Earth is $R_e$]

Imagine a rocket with mass $M$ is moving toward a gas cloud with initial velocity $v$. Naturally the rocket will be colliding with the gas particles and gradually losing its kinetic energy. Now for the sake of simplicity, we can model this gas cloud as tiny particles with very small mass $m$ and density $N$ $particles/m^3$ The cross sectional area of the rocket is $A$. Now find out how much time it will take to decrease the rocket’s velocity to half of its original.

Litu is trying to balance one egg on another egg, which rests on a table. He does it by taking an empty cylindrical plastic cup. The cup is inverted and put over the two eggs, so that the inner wall of the cup prevents the upper egg from falling down on the table.

Figure out the condition for the mass of the cup in terms of mass of eggs, radius of egg(assume eggs are spherical) and cup so that the cup tumbles over.

Many particle physics experiments are carried on one of two types of accelerator facilities, (a) colliders in which equal energy particle collide head on, i.e. their momentum vectors point in opposite direction, (b) “fixed target facilities”, where an energetic beam particle collide with a stationary target particle. The two types of facilities have different advantages and

disadvantages. We’ll consider one important difference in this problem.

Suppose our goal is to collide two particles of same mass m to create a more massive particle of mass $M$ $(M > m)$ – in other words the original particles of mass m annihilate with each other and create a new particle of mass $M$. The key question regarding how much expensive our particle accelerator will be “How much energy does our particle beam(s) need to have?’’

(a)

(b)

(c) What is the ratio of the fixed target beam and the sum of the energies of two target beams in head on collisions? Which facility do you think should be less expensive?

What is this ratio when $m$ is the mass of the proton $(~ 1 GeV/c2)$ and $M$ is the mass of the Higgs Boson $(~125 GeV/c2)$

*$Problem-1$*A coherent beam of monochromatic light of wavelength $500$ $nm$ is directed towards two thin slits separated by a distance of $1$ $mm$. If the screen covers an angular region of $60˚$, as shown in the figure, approximately how many bright interference bands are visible on the screen?

*$Problem-2$*Consider a conducting sphere with charge $+q$ and a point charge $-q$ kept as shown in the figure. Draw electric field lines.

*$Problem-3$*A large loop of radius $R$ carries a current of $I$. A much smaller loop of radius $r$ is placed at the center of this loop, such that its axis is perpendicular to that of the larger loop. The smaller loop carries a much smaller current $i$. What is the torque on the smaller loop?

*$Problem-4$*Assume we have a long column of water and an air bubble of radius $r$ and pressure $P_0$ at the surface of the Earth is rising inside that. If the column is long enough for $g$ not to be considered constant, find $r$ as a function of $R$; where $R$ is the distance between the bubble and the center of Earth. [The radius of Earth is $R_e$]

*$Problem-5$*Imagine a rocket with mass $M$ is moving toward a gas cloud with initial velocity $v$. Naturally the rocket will be colliding with the gas particles and gradually losing its kinetic energy. Now for the sake of simplicity, we can model this gas cloud as tiny particles with very small mass $m$ and density $N$ $particles/m^3$ The cross sectional area of the rocket is $A$. Now find out how much time it will take to decrease the rocket’s velocity to half of its original.

*$Problem-6$*Litu is trying to balance one egg on another egg, which rests on a table. He does it by taking an empty cylindrical plastic cup. The cup is inverted and put over the two eggs, so that the inner wall of the cup prevents the upper egg from falling down on the table.

Figure out the condition for the mass of the cup in terms of mass of eggs, radius of egg(assume eggs are spherical) and cup so that the cup tumbles over.

*$BONUS$***At the Large Hadron Collider**Many particle physics experiments are carried on one of two types of accelerator facilities, (a) colliders in which equal energy particle collide head on, i.e. their momentum vectors point in opposite direction, (b) “fixed target facilities”, where an energetic beam particle collide with a stationary target particle. The two types of facilities have different advantages and

disadvantages. We’ll consider one important difference in this problem.

Suppose our goal is to collide two particles of same mass m to create a more massive particle of mass $M$ $(M > m)$ – in other words the original particles of mass m annihilate with each other and create a new particle of mass $M$. The key question regarding how much expensive our particle accelerator will be “How much energy does our particle beam(s) need to have?’’

(a)

**(Head on)**What is the minimum relativistic total energy of each particle of mass $m$ must be to create the new particle with mass $M$? (Express your answer in terms of $m$ and $M$)(b)

**(Fixed Particle)**What is the minimum total energy that our incident particle of mass $m$ must have if it is to collide with another stationary particle of mass $m$ and create the single particle with mass $M$? Express your answer in terms of $m$ and $M$. Hint: Don’t forget that momentum must be conserved!(c) What is the ratio of the fixed target beam and the sum of the energies of two target beams in head on collisions? Which facility do you think should be less expensive?

What is this ratio when $m$ is the mass of the proton $(~ 1 GeV/c2)$ and $M$ is the mass of the Higgs Boson $(~125 GeV/c2)$