The bent wire shown in Fig. 28 42 lies in a uniform magnetic...

The bent wire shown in Fig. 28 42 lies in a uniform magnetic field. Each straight section is $2.0 \mathrm{~m}$ long and makes an angle of $\theta=60^{\circ}$ with the $x$ axis, and the wire carries a current of $2.0 \mathrm{~A}$. What is the net $\mathrm{mag}$ netic force on the wire in unit-vector notation if the magnetic field is given by (a) 4.0k T and (b) 4.0\hat T?

Key Concepts

Vector Addition of Forces

When a conductor is composed of multiple segments or parts, each segment can experience a distinct magnetic force. The overall net force on the conductor is then determined by the vector sum of these individual forces, requiring careful addition of both magnitude and directional components.

Unit Vector Notation

Unit vector notation is a systematic method for representing vector quantities, such as force, in terms of their components along standard coordinate axes. This notation ensures clarity when conveying both the magnitude and directional aspects of a vector in a concise and standardized form.

Uniform Magnetic Field

A uniform magnetic field is one where the magnetic field has a constant magnitude and direction throughout the region of interest. In problems involving conductors, this assumption simplifies calculations as each segment of the wire is subjected to the same magnetic field properties.

Lorentz Force Law

This law establishes that a current-carrying wire located in a magnetic field experiences a force given by F = I (L × B), where I is the current, L is the length vector of the wire, and B is the magnetic field. It encapsulates how the direction and magnitude of a magnetic force depend on the interplay between the current direction and the orientation of the magnetic field.

Vector Cross Product

The cross product is a vector operation used to compute the force on the wire by taking the product of the current element (the length vector) and the magnetic field vector. It inherently produces a vector that is perpendicular to both contributing vectors, aligning with the physics described by the right?hand rule.

Transcript

00:01 In this question, we have this band's wire.

00:06 Lines in the uniform magnetic field.

00:08 It has a section.

00:10 The length is 2 meters for each section.

00:14 And then the current is 2 a pampus.

00:20 So we want to find a net magnetic force on the wire when the magnetic field is 4 tesla in the k -head direction and 4 tesla in the i -head direction.

00:31 Okay, so to do this question, we'll be using the formula fb equals to il cross b so i'm going to call this wire 1 this wire 2 and then the angle data is 60 degrees so the vector l is the direction of currents in the wire and for wire 1 l is 2 xx0x0 x0 degrees i head plus sine 60 degrees j head so if you evaluate you give you i head plus square of 3 j hat and then l2 is 2 times cosine 60 degrees i hat minus sine 60 degrees j head so you get i hat minus square of 3 j head so you get i hat minus square of 3 j head so um and in part our b is in the k head direction okay so we can find the force on wire 1 would be i times l1 cross b so we have the current is 2 ampiers then i had plus 32 j head plus 3 j head cross 4 k hat all right so get 8 times so i cross k get negative j hat j cross k get i head so you have square root three i head minus j hat nymphins okay and then for the force on wire two you repeat the same formula and then so you have two times i head minus square three j hat cross four k hat then you are going to get 8 times negative root 3 ihad, negative j hat, newtons.

02:52 Okay? so the next force will just be f1 plus f2.

03:01 You can see that the ihad component can start out, you are left with negative 16 j hat, newtons...

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