A magnetic dipole with a dipole moment of magnitude 0.020 J / T is released from rest in a unif

A magnetic dipole with a dipole moment of magnitude 0.020 J...

A magnetic dipole with a dipole moment of magnitude $0.020 \mathrm{~J} / \mathrm{T}$ is released from rest in a uniform magnetic field of magnitude $46 \mathrm{mT}$. The rotation of the dipole due to the magnetic force on it is unimpeded. When the dipole rotates through the orientation where its dipole moment is aligned with the magnetic field, its kinetic energy is $0.80 \mathrm{~mJ}$. (a) What is the initial angle between the dipole moment and the magnetic field? (b) What is the angle when the dipole is next (momentarily) at rest?

Key Concepts

Torque on a Magnetic Dipole

The torque experienced by a magnetic dipole in a magnetic field, given by ? = ?B sin?, causes the dipole to rotate so as to align itself with the magnetic field. This torque is responsible for changing the potential energy and, subsequently, the kinetic energy of the system as the dipole moves from one orientation to another.

Conservation of Energy

The principle of conservation of energy is applied to analyze the rotational motion of a magnetic dipole in a magnetic field. As the dipole rotates, any decrease in magnetic potential energy is converted into kinetic energy, allowing one to relate changes in orientation to the measured kinetic energy, assuming there are no energy losses due to friction or other non-conservative effects.

Magnetic Potential Energy

When a magnetic dipole is placed in a magnetic field, it possesses potential energy given by the expression U = -?B cos?, where ? is the dipole moment, B is the magnetic field strength, and ? is the angle between the dipole moment and the field direction. This concept is crucial for understanding how energy is exchanged between potential and kinetic forms as the dipole rotates.

Magnetic Dipole Moment

The magnetic dipole moment is a vector quantity that represents the strength and orientation of a magnet's dipole. It determines how the dipole interacts with external magnetic fields, including the magnitude of the torque it experiences and the potential energy associated with its orientation in the field.

Transcript

00:01 Hello student in this question we have given a dipole moment value that is we can denote by p that is equals to 0 .02 zh tzla so this is our given values we can write this these are our given values first and second value that is agnetic field that is equal to 46 milli tesla and the third value this is that is given in the cushion that is kinetic energy its value is 0 .8 bly zool so basically in this question we are having a dipole moment p and it's making an angle with magnetic field that we know we have to calculate in first part theta so in this question and this initial value of kinetic energy is zero so when b initial is 0 to 0 the final there will be some kinetic energy and here the maximum potential energy the potential energy will be maximum so for this first part we can use the energy conservation process energy conservation so here we know that kinetic energy plus potential energy that is initial value is equal to kinetic final energy plus potential final energy plus potential final energy so initial kinetic energy that is zero plus the potential energy in the magnetic field that is minus of m dot b and equals to the kinetic energy of the final value this is final value of kinetic energy that is given 0 .8 multiply by 10 to power minus 3 because this is millijoule so this is jule plus the potential energy that is minus m that is minus m into or dot we can say dot bose 0 0 0 0 0 0 0 0.

02:23 So after this dot we can open like minus m into b course theta that we have to calculate the angle between b and v so after putting all these values we can see that we can say that the kinetic energy is zero and potential energy is after putting these value and this is this value is equal to 1 so after rearranging these equation we get mb 1 minus 4 theta is the angle between the dipole moment and the magnetic fade is equal to 0 .8 multiply by 10 raise to 4 minus 3 jule or we can jule here we can say this so after putting these values we get 1 minus 0x theta multiply by 0 .02 multiply by magnetic field that is 46 divided by thousand is equal to 0 .8 divided by thousand so here we can cancel these two values that is thousand and after calculating rearranging these values by this dividing 0 .8 by 0 .02 and multiplied by 46 so we get here 1 minus 4 theta 1 minus 4 theta is equal to 80 divided by 92 and by regenerating by transferring these values in that side we get 4 theta is equal to 12 by 12 divided by 92 or we can say the final answer is equal to pose theta theta is the angle between these two values these are my dipole and we get 6 by 46 or we can you can further calculate these values so this is our answer of first part answer of first part and in second part we have to in second part, see in second part there is a situation that is if a dipole moment is making an angle with b, some angle theta, that we have calculated before that is 6 by 46...

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