A loudspeaker creates sound by pushing air back and forth...

A loudspeaker creates sound by pushing air back and forth with a paper cone that is driven by a magnetic force on a wire coil at the base of the cone. Figure $\mathrm{P} 24.52$ shows the details. The cone is attached to a coil of wire that sits in the gap between the poles of a circular magnet. The 0.18 T magnetic field, which points radially outward from $\mathrm{N}$ to $\mathrm{S}$, exerts a force on a current in the wire,moving the cone. The coil of wire that sits in this gap has a diameter of $5.0 \mathrm{cm},$ contains 20 turns of wire, and has a resistance of $8.0 \Omega .$ The speaker is connected to an amplifier whose instantaneous output voltage of $6.0 \mathrm{V}$ creates a clockwise current in the coil as seen from above. What is the magnetic force on the coil at this instant?

Key Concepts

Coil Geometry and Multiple Turns

The geometry and configuration of a coil, including its diameter and the number of turns, significantly affect its interaction with a magnetic field. Multiple loops or turns in a coil mean that the total force or other electromagnetic effects are the cumulative result of the contributions from each segment of the coil. This concept is critical when designing components that rely on electromagnetic forces for functionality.

Magnetic Field

A magnetic field is a vector field that exerts magnetic forces on moving charges and current-carrying wires. Its strength and direction at any point in space determine the magnitude and orientation of the force acting on conductors. Magnetic fields play a key role in the operation of various electromechanical devices and systems.

Lorentz Force on a Current-Carrying Conductor

This concept explains that a conductor carrying an electric current experiences a force when placed in an external magnetic field. The force is given by the vector cross product of the current element and the magnetic field, highlighting both the magnitude and direction—typically determined by the right-hand rule. This principle is crucial in understanding how devices like loudspeakers and motors convert electrical energy into mechanical motion.

Ohm's Law

Ohm's Law relates the voltage applied across a conductor to the current flowing through it and the resistance of the conductor. It provides the fundamental relationship V = IR, which is essential for determining the current in any circuit. This current value is often needed when calculating other quantities, such as the magnetic force acting on a conductor in a magnetic field.

Transcript

00:01 Okay, so for this question, window b1 can be equal to the x component of b1 plus the y component of b1, which is b1xi plus b1yj.

00:10 Since both components is along the positive direction, so both are positive, and they can be also equal to b sine 60i plus b cosine 60 degree j, which is equal to b tens square three i plus b tens 1 half j.

00:31 Eventually we'll have b1 just equal to b times square root 3 over 2 times i plus 1 half j.

00:37 And for b2 we have two components as well, but the x component is negative.

00:44 Okay, so we have negative b2 x i plus b2y j.

00:48 Eventually we have b2 is equal to b times negative square root 3 over 2 i plus 1 half j.

00:57 So we can determine f1 now, okay, which is the force vector.

01:01 And this is equal to ib1 cross product with l1.

01:05 L1 is the length of the loop.

01:08 Okay, so since it's cross product, so that means, and we know that l1 can be equal to l times negative k in this case.

01:18 So eventually we have f1 vector is equal to ib vector times square three over two i plus one half j, and then cross product with l times negative k.

01:29 And since it's cross product, we know i cross product with k is j, j cross product with k is i.

01:36 So eventually we have f1 vector, it's good to i b times negative square three over two times j minus 1 half i times l vector.

01:46 And f2, which is the force 2 vector, is equal to ib2 cross product with the l2, which is ib times negative 3 over 2 i plus 1 1 .5 j, and then cross product with lk.

02:02 So eventually you have f2 is equal to ib tends negative square of 3 over 2 j plus 1 half i times vector l...

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