A stationary circular wall clock has a face with a radius of...

A stationary circular wall clock has a face with a radius of 15 cm. Six turns of wire are wound around its perimeter; the wire carries a current of 2.0 $\mathrm{A}$ in the clockwise direction. The clock is located where there is a constant, uniform external magnetic field of magnitude 70 $\mathrm{mT}$ (but the clock still keeps perfect time). At exactly $1 : 00 \mathrm{eM},$ the hour hand of the clock points in the direction of the external magnetic field. (a) After how many minutes will the minute hand point in the direction of the torque on the winding due to the magnetic field? (b) Find the torque magnitude.

A stationary circular wall clock has a face with a radius of 15 cm. Six turns of wire are wound around its perimeter; the wire carries a current of 2.0 $\mathrm{A}$ in the clockwise direction. The clock is
located where there is a constant, uniform external magnetic field of magnitude 70 $\mathrm{mT}$ (but the clock still keeps perfect time). At
exactly $1 : 00 \mathrm{eM},$ the hour hand of the clock points in the direction
of the external magnetic field. (a) After how many minutes will the minute hand point in the direction of the torque on the winding
due to the magnetic field? (b) Find the torque magnitude.

Key Concepts

Rotational Kinematics for Clock Hands

The analysis of clock hands involves converting angular displacement into time, using the fact that the hands move with constant angular speed. A complete revolution (360°) occurs in a fixed period (60 minutes for the minute hand), so any angular displacement can directly be related to the corresponding elapsed time. This concept allows one to determine the time interval corresponding to a specific angular difference between directions on the clock face.

Magnetic Dipole Moment

A current-carrying loop (or a series of loops) behaves like a magnetic dipole, characterized by a dipole moment (?) that is proportional to the current, the number of turns, and the area enclosed by the loop. Its magnitude is given by ? = N·I·A, and its direction is determined by the right?hand rule. This concept is fundamental in understanding how currents interact with magnetic fields.

Torque on a Magnetic Dipole in a Uniform Magnetic Field

When a magnetic dipole is placed in a uniform magnetic field, it experiences a torque that tends to align the dipole moment with the field. The torque has a magnitude ? = ?B sin?, where ? is the angle between the magnetic dipole moment and the magnetic field. This cross product relation explains why the dipole experiences its maximum torque when it is perpendicular to the magnetic field.

Transcript

00:01 Hello student this is a problem of magnetic field and in this problem we have given a stationary circular wall clock and it has a radius of 15 cm and 6 tons of wire are around and you will write n is equal to 6 the wire carries a current 2 ampere in clockwise direction so the magnetic field is equal to 70 milt tesla and it is towards at one the hour end of the clock points in the direction of the external magnetic field so we have to find after how many minutes will the minute hand point in the direction of the torque on the winding due to the magnetic field so suppose this is a diagram of clock and at 1 o 'clock the hour end will be in this direction.

01:25 That means this is the direction of magnetic field.

01:30 And minute end will be here at 1 o 'clock.

01:36 Now the current is moving in clockwise direction.

01:40 That means the dipole movement will be in into the page in this direction.

01:48 Here.

01:49 This is the direction of dipole movement because the current in clockwise direction so by right -hand rule we can get the direction of magnetic field which is into the page so we know that the direction of torque is equal to torque is equal to cross -product of dipole moment and magnetic field this means the torque should be perpendicular to the torque should be perpendicular to the magnetic field and as well as dipole movement so here this should be the direction of sorry from perpendicular to the magnetic field and the dipole moment so this should be the direction of torque now if we see this clock this is one this is two this is three and this is four five six and here so if we see the clock diagram, we can clearly say if after 20 minutes, the minute hand will be at 4 and it is making 90 degree angle with magnetic field.

03:13 So we can write t equal to 20 minutes...

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