A Voice Coil. It was shown in Section 27.7 that the net...

A Voice Coil. It was shown in Section 27.7 that the net force on a current loop in a uniform magnetic field is zero. The magnetic force on the voice coil of a loudspeaker (see Fig. 27.28 ) is nonzero because the magnetic field at the coil is not uniform. A voice coil in a loudspeaker has 50 turns of wire and a diameter of $1.56 \mathrm { cm } ,$ and the current in the coil is 0.950 A. Assume that the magnetic field at each point of the coil has a con- stant magnitude of 0.220$\mathrm { T }$ and is directed at an angle of $60.0 ^ { \circ }$ outward from the normal to the plane of the coil (Fig. $\mathrm { Pa } 7.83 )$ . Let the axis of the coil be in the $y$ -direction. The current in the coil is in the direction shown (counterclockwise as viewed from a point above the coil on the $y$ -axis). Calculate the magnitude and direction of the net magnetic force on the coil.

A Voice Coil. It was shown in Section 27.7 that the
net force on a current loop in a
uniform magnetic field is zero. The
magnetic force on the voice coil of a
loudspeaker (see Fig. 27.28 ) is nonzero because the magnetic field
at the coil is not uniform. A voice
coil in a loudspeaker has 50 turns of
wire and a diameter of $1.56 \mathrm { cm } ,$ and
the current in the coil is 0.950 A. Assume that the magnetic field at each point of the coil has a con-
stant magnitude of 0.220$\mathrm { T }$ and is directed at an angle of $60.0 ^ { \circ }$ outward from the normal to the plane of the coil (Fig. $\mathrm { Pa } 7.83 )$ . Let the axis of the coil be in the $y$ -direction. The current in the coil is in
the direction shown (counterclockwise as viewed from a point above
the coil on the $y$ -axis). Calculate the magnitude and direction of the
net magnetic force on the coil.

Key Concepts

Right-Hand Rule and Vector Cross Product

The right-hand rule is an essential tool for determining the direction of the magnetic force experienced by a current element in a magnetic field. It is directly related to the vector cross product used in the force equation, and helps in visualizing and predicting the orientation of forces in three dimensions, which is critical for the design and analysis of electromagnetic devices.

Magnetic Field Uniformity and Its Effects

Uniformity of the magnetic field is key to predicting the behavior of current loops. In a uniform field, symmetry ensures that forces on opposite segments of a loop cancel. In contrast, non-uniform fields or fields with directional variation can produce a net force or torque on the loop, which is the operating principle behind many electromagnetic actuators and voice coils.

Integration of Forces in a Closed Loop

For a closed current loop, individual force contributions on each segment of the wire may cancel out in a uniform magnetic field, resulting in a zero net force. However, if the magnetic field is non-uniform or oriented at an angle relative to the loop, the cancellation is incomplete and a net force can arise. This principle is vital in understanding how electrical energy is converted to mechanical motion in systems like loudspeakers.

Magnetic Force on a Current-Carrying Conductor

This concept explains that a segment of current-carrying wire in a magnetic field experiences a force given by the expression F = I (L × B), where I is the current, L is the length vector of the wire segment, and B is the magnetic field. The direction of the force is determined by the cross product, which plays a crucial role in understanding the interaction between current and magnetic fields in electromagnetic devices.

Transcript

00:01 So we are going to use df equal to i times dl times b sine 5 to calculate the force on a short segment of the coil.

00:14 And then we are going to integrate over the entire coil to find the total force.

00:19 Now let's take a look at these two figures.

00:24 So these are basically sketches that mark the direction of current magnetic field and the force for two short segments of the coil on opposite sides of the coil.

00:45 So you can see that from these two sketches that the x components cancel because i and b are are constant in these two.

00:56 So the magnitude of dfx and dfx dash are equal, but they are in opposite direction.

01:04 So for this part, the net force, which is going to be some of these two forces, is going to be zero.

01:11 And so the net force along y direction, so the net force is going to be the sum of the forces along y direction.

01:22 So only the y components will add.

01:25 And this is true for all pairs of short segments on opposite sides of the coil.

01:32 That means that the net magnetic force on the coil is in the wide direction because the x components simply cancel and its magnitude will be given by f, let's say, and this will be given by integration over all components along wide direction.

01:58 Now let's calculate that.

02:01 So df, which is net force for, let's say the short segment one, is equal to i times d l times b, sine phi, as i wrote here.

02:19 And phi is 90 degree because current and magnetic field are perpendicular.

02:24 So this will just be equal to i times b times d l.

02:29 So dfy will be equal to df times cost 30 degree...

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