A long wire carrying 4.50 A of current makes two 90^∘ bends, as shown in Fig. E 27.39 . The bent

step 1 Okay, so let's see if we get this one figured out. So we're given that we have a long wire carry

A long wire carrying 4.50 A of current makes two 90^∘ bends,...

A long wire carrying 4.50 A of current makes two $90^{\circ}$ bends, as shown in Fig. $\mathrm{E} 27.39$ . The bent part of the wire passes through a uniform $0.240-\mathrm{T}$ magnetic field directed as shown in the figure and confined to a limited region of space. Find the magnitude and direction of the force that the magnetic field exerts on the wire.

A long wire carrying 4.50 A of current makes two $90^{\circ}$ bends, as shown in Fig. $\mathrm{E} 27.39$ . The bent part of the wire passes through a uniform $0.240-\mathrm{T}$ magnetic field directed as shown in the figure and confined to a limited region of space. Find the
magnitude and direction of the force that the magnetic field exerts on the wire.

Key Concepts

Magnetic Force on a Current-Carrying Conductor

This concept describes how a current-carrying conductor experiences a force when placed in a magnetic field. The force is given by the Lorentz force law, expressed as F = I (L × B), where I is the current, L is the vector representation of the length and direction of the conductor, and B is the magnetic field. The force acts perpendicular to both the current direction and the magnetic field, and its magnitude and direction depend on the relative orientation of these vectors.

Right-Hand Rule

The right-hand rule is a practical method for determining the direction of the magnetic force on a current-carrying wire. By orienting the thumb in the direction of the current and the fingers in the direction of the magnetic field, the direction in which the palm pushes indicates the direction of the magnetic force. This rule is essential for visualizing and predicting the vector nature of the force created by the interaction between current and a magnetic field.

Vector Addition of Forces

When a conductor is composed of multiple segments or curves, each segment may experience a magnetic force in a different direction. The net force on the conductor is determined by the vector sum of these individual forces. Understanding how to add forces vectorially is crucial when analyzing systems where the geometry of the current path influences the overall magnetic interaction.

Integration of Differential Forces

For a non-uniform or curved current path, the total magnetic force is found by integrating the differential force contributions along the conductor. This is done using the expression dF = I (dl × B), where dl is an infinitesimal segment of the conductor. This integration accounts for varying orientations and lengths along the wire, enabling an accurate calculation of the total force exerted by the magnetic field.

Transcript

00:01 Okay, so let's see if we get this one figured out.

00:03 So we're given that we have a long wire carrying a 4 .50 ampere of current.

00:10 It makes two 90 -degree bends as shown.

00:13 The bent part of the wire passes through a uniform 0 .240 t magnetic field.

00:21 It's confined to a limited region in space.

00:24 We're asked to find the magnitude and direction of the force the magnetic field exerts on the wire.

00:33 Okay, so we can see that we have a magnitude and b equals 0 .240t, or la, we have is equal to x, or lc is equal to 60 centimeters, minus x, or lb is 30 centimeters, current is 4 .50 amps.

01:10 The force acting on the first and third are in the same direction, downward.

01:17 The other one is to the right.

01:18 The direction is 90 degrees.

01:22 Okay, so then we can use, oops, we can write, and this will be, okay, and we'll add, okay, i'm figuring this out.

02:50 So i believe this will be 0 .648 newtons.

02:57 This is my vertical force.

03:08 Then let's do our horizontal force, and that'll be 450, 4 .50 times 0 .30 times 0 .240, and that will be 0 .340.

03:40 Okay, so the magnitude of my force, and that will be, my first one was 0 .324, or that's my second one, 0 .648.

04:13 Okay...

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