A conducting coil consisting of n turns of wire is placed in...

A conducting coil consisting of $n$ turns of wire is placed in a uniform magnetic field given by $\vec{B}=2.00 \hat{y} \mathrm{~T},$ as shown in the figure. The radius of the coil is $R=5.00 \mathrm{~cm},$ and the angle between the magnetic field vector and the unit normal vector to the coil is $\theta=60.0^{\circ} .$ The current through the coil is $i=5.00 \mathrm{~A}$. a) Specify the direction of the current in the coil, given the direction of the magnetic dipole moment, $\vec{\mu},$ shown in the figure. b) Calculate the number of turns, $n$, the coil must have for the torque on the loop to be $3.40 \mathrm{~N} \mathrm{~m}$ c) If the radius of the loop is decreased to $R=2.50 \mathrm{~cm}$ what should the number of turns, $N$, be for the torque to remain unchanged? Assume that $i, B,$ and $\theta$ stay the same.

A conducting coil consisting of $n$ turns of wire is placed in a uniform magnetic field given by $\vec{B}=2.00 \hat{y} \mathrm{~T},$ as shown in the figure. The radius of the coil is $R=5.00 \mathrm{~cm},$ and the angle between the magnetic field vector and the unit normal vector to the coil is $\theta=60.0^{\circ} .$ The current through the coil is $i=5.00 \mathrm{~A}$.
a) Specify the direction of the current in the coil,
given the direction of the magnetic dipole moment,
$\vec{\mu},$ shown in the figure.
b) Calculate the number of turns, $n$, the coil must have for the torque on the loop to be $3.40 \mathrm{~N} \mathrm{~m}$
c) If the radius of the loop is decreased to $R=2.50 \mathrm{~cm}$ what should the number of turns, $N$, be for the torque to remain unchanged? Assume that $i, B,$ and $\theta$ stay the same.

Key Concepts

Magnetic Dipole Moment

This concept refers to the vector quantity produced by a current loop, given by ? = i • n • A, where i is the current, n is the number of turns in the coil, and A is the area of the loop. It represents the magnetic strength and orientation of the coil and determines how the coil interacts with an external magnetic field.

Torque on a Magnetic Dipole

When a magnetic dipole is placed in a uniform magnetic field, it experiences a torque given by ? = ? × B. This torque tends to align the magnetic dipole moment with the direction of the magnetic field. The magnitude of the torque depends on the magnitudes of ? and B, as well as the sine of the angle between them.

Coil Area and Its Role in Magnetic Effects

The area of a loop, often calculated as A = ?R² for a circular coil, directly influences the magnetic dipole moment. In the context of a coil in a magnetic field, changes in the radius alter the area, thereby affecting the dipole moment and ultimately the torque experienced by the coil.

Right-Hand Rule

This rule is used to determine the direction of the magnetic dipole moment relative to the current flow in the coil. By curling the fingers of the right hand in the direction of current circulation, the thumb points in the direction of the dipole moment. This helps in predicting the orientation of the torque in the magnetic field.

Transcript

00:02 Hi, in the given problem here, these are the two pole pieces, the north pole and the south pole, which are creating this magnetic field like this, from north to south.

00:18 This is n, this is s.

00:21 In the aerial view, in the top view of this diagram, here this is the plane of the coil, the circular coil.

00:28 This is its magnetic moment coming out of its plane which is visible to us.

00:35 This is the symbol for the magnetic moment.

00:38 It is a vector quantity and the magnetic moment is making an angle of 60 degree with the direction of magnetic field, b bar.

00:48 Here this is the diameter of the coil to r.

00:54 The magnitude, the complete value of the magnetic field in vector form is given as 2 .00 y cap tesla means this is along y -axis.

01:09 Radius of the coil is given as r is equal to 5 .00 centimeter angle between the normal drawn at the plane of coil and the magnetic field is 60 degree.

01:28 Current passing through the coil is 5 .00 ampere.

01:32 Now in the first part of the problem we have to find the direction of current passing through this loop.

01:37 Now as the magnetic moment of the coil means mu bar is coming out of the plane of the coil, plane of the coil which is visible to us and we know magnetic moment is directed always from south to north hence it is clear that the upper face of this coil which is visible to us through which the magnetic moment is coming out is north pole it is behaving like north pole so the direction of current in it is counter clock wise which becomes the answer for the first part of the problem now in the second part of the problem torque acting on this coil is given as 3 .40 newton into meter but we know an expression for the magnetic torque acting on the current carrying coil which is suspended in a magnetic field.

03:31 That expression is given as the product of magnetic field with the current passing through the coil with the number of turns with the area of the coil and sine theta where theta is the angle between the plane, between the normal drone at the plane of the coil and the magnetic field.

03:48 So plugging in all known values here, this is 3 .40 newton into meter for the torque.

03:54 For magnetic field, this is 2 tesla for current v...

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