A straight wire is aligned north-south in a region where...

A straight wire is aligned north-south in a region where Earth's magnetic field $\overrightarrow{\mathbf{B}}$ is directed $58.0^{\circ}$ above the horizontal, with the horizontal component directed due north. The wire carries a current of $8.00 \mathrm{A}$ toward the south. The magnetic force on the wire per unit length of wire has magnitude $2.80 \times 10^{-3} \mathrm{N} / \mathrm{m} .$ (a) What is the direction of the magnetic force on the wire? (b) What is the magnitude of $\overline{\mathbf{B}} ?$

A straight wire is aligned north-south in a region where Earth's magnetic field $\overrightarrow{\mathbf{B}}$ is directed $58.0^{\circ}$ above the horizontal, with the horizontal component directed due north. The wire carries a current of $8.00 \mathrm{A}$ toward the south. The magnetic force on the wire per unit length of wire has magnitude $2.80 \times 10^{-3} \mathrm{N} / \mathrm{m} .$
(a) What is the direction of the magnetic force on the wire? (b) What is the magnitude of $\overline{\mathbf{B}} ?$

Key Concepts

Decomposition of Magnetic Fields

This concept involves breaking a magnetic field vector into its horizontal and vertical components, which is particularly useful when the magnetic field is not aligned with the standard axes. In many real-world problems, such as those involving Earth’s magnetic field, this decomposition helps in analyzing the effects of the field on objects oriented in a specific direction.

Vector Cross Product

The vector cross product is a mathematical operation that takes two vectors and produces a third vector perpendicular to the plane containing the initial vectors. In the context of magnetic forces, it is used to compute the magnitude and direction of the force on a current-carrying conductor via F = I L x B, where the sine of the angle between L and B determines the magnitude. This operation is key in understanding the directional properties of electromagnetic interactions.

Force per Unit Length

Force per unit length is a measure that describes how the total force experienced by an object, like a wire, is distributed along its length. It is especially useful in electromagnetic problems for simplifying calculations, as it allows one to relate the force acting on a small segment of the conductor to the overall force, using the formula F/L = I B sin?.

Magnetic Force on a Current-Carrying Wire

This concept describes how a conductor that carries an electric current experiences a force when placed in a magnetic field. The force is given by the relation F = I L x B, where I is the current, L is the length vector of the wire, and B is the magnetic field. Understanding this interaction is fundamental in electromagnetism and has practical applications in devices like electric motors and generators.

Right-Hand Rule

The right-hand rule is a mnemonic used to determine the direction of the force acting on a current-carrying wire in a magnetic field. By pointing the thumb in the direction of the current and the fingers in the direction of the magnetic field, the palm faces the direction of the force. This method is crucial for visualizing the vector cross product that governs the interaction between the current and the magnetic field.

Transcript

00:01 In the given problem here, we are representing north direction.

00:10 So if this is north, this is south, this will be south, then here it will be east and it will be west here.

00:22 Now the magnetic field is outward, that is coming out of the earth surface like this and making an angle theta with the horizontal this angle theta is given as 58 .0 degree.

00:46 So this magnetic field will be having two components.

00:49 One horizontal and another component will be coming out of the plane of this paper.

00:56 So we represent it by a dot.

00:59 The current passing through the wire and that is due south.

01:06 The wire has been kept along north -south.

01:12 This is the wire and this is the current passing through it.

01:17 Magnitude of this current is 8 .00 ampere due south.

01:28 Now earth's magnetic field is having one horizontal component, along south to north so the conductor will not experience any force any magnetic force due to this horizontal component being antiparralralral angle between them will be 180 degree so, sine 180 degree we know it will come out to be 0.

02:47 That's why the force, magnetic force will also become 0.

02:51 And another component is vertically up.

03:08 And that component will be given by bv is equal to b.

03:15 Sin delta.

03:17 And the conductor will experience a magnetic force only due to this component...

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