In Fig. 29-52, a current i=2.2 A is set up in a long hairpin conductor formed by bending a wire

In Fig. 29-52, a current i=2.2 A is set up in a long...

In Fig. 29-52, a current $i=2.2 \mathrm{~A}$ is set up in a long hairpin conductor formed by bending a wire into a semicircle of radius $R=8.5 \mathrm{~mm} .$ Point $b$ is midway between the straight sections and so distant from the semicircle that each straight section can be approximated as being an infinite wire. What are the (a) magnitude and (b) direction (into or out of the page) of $\vec{B}$ at $a$ and the (c) magnitude and (d) direction of $\vec{B}$ at $b$ ?

In Fig. 29-52, a current $i=2.2 \mathrm{~A}$
is set up in a long hairpin conductor
formed by bending a wire into a
semicircle of radius $R=8.5 \mathrm{~mm} .$
Point $b$ is midway between the
straight sections and so distant from the semicircle that each
straight section can be approximated as being an infinite wire.
What are the (a) magnitude and (b) direction (into or out of the
page) of $\vec{B}$ at $a$ and the (c) magnitude and (d) direction of
$\vec{B}$ at $b$ ?

Key Concepts

Right-Hand Rule

The right-hand rule is a mnemonic used to determine the direction of the magnetic field generated by a current. By pointing the thumb of the right hand in the direction of conventional current flow, the curled fingers indicate the direction of the magnetic field lines around the conductor. This rule is vital for understanding the orientation of fields in various current configurations.

Superposition Principle in Electromagnetism

The superposition principle states that when multiple magnetic field sources are present, the net magnetic field at any point is the vector sum of the individual fields produced by each source. This principle is key in solving complex problems where currents flow in different segments (curved and straight), allowing one to calculate the overall magnetic field by considering one contribution at a time.

Magnetic Field of a Circular Arc

The magnetic field produced by a circular arc is derived from the Biot–Savart law, where the contribution of each differential segment of the arc is integrated over its extent. The field at the center (or edge) of the circular segment depends on the amount of the circle, usually expressed in terms of the angle spanned by the arc, and features a characteristic dependence on the radius. This concept is particularly important when analyzing current distributions involving partial loops or curved segments.

Biot–Savart Law

The Biot–Savart law is a fundamental principle used to calculate the magnetic field produced by a small segment of current. It states that the field produced is directly proportional to the current element and inversely proportional to the square of the distance to the point where the field is being calculated. This law is essential for determining the magnetic field distribution in systems with known current paths.

Magnetic Field due to a Straight Current-Carrying Wire

The magnetic field generated by an infinite straight wire is given by a straightforward expression derived from Ampère’s law or the Biot–Savart law. The magnitude of the field at a point is inversely proportional to the perpendicular distance from the wire. This concept is widely used in magnetic field calculations involving long, straight conductors, where the simplicity of the formula facilitates easy computation.

Transcript

00:01 Currents i, 2 .2 ampere, is set in a long hairpin conductor found by bending a wire into semicircle of radius 8 .5 m m.

00:11 Point b is midway between the straight sections and so distant from the semicircle that each straight section can be approximated as infinite wire.

00:21 Then what are the magnitude and direction of the magnetic field at 8? and magnitude and direction of the magnetic field at b magnetic field at point a consists of the magnetic field due to semicircle and due to semi -infinite wire for semicircle it is nu not i pi by 4 pi r which is equal to new not i by 4rr now this infatimal magnetic field for semi -infinite equals mu not i d x i cross r cap divided by 4 pi into r square this will be equals to nu not i r by 4 pi into d x by r squared plus x squared raised to the power 3 by k cap.

01:42 The value of x ranges from 0 to infinite therefore integrating it within the limits of 0 to infinite we get the magnetic field for the sine infinite is nu not i r by 4 5 integrating from 0 to infinite d x by r square plus x squared raised to the power 3 by 2 k cap here x equals r tan theta then d x will be equals to r s square theta d theta then r sq square square plus x square can be written as r squared sec square theta using this into the equation we have the integral of d x by r square plus x square raised power 3 by 2 k cap equals 1 by r square x divided by root of x square plus r square k -capped plus constant.

02:58 Evaluating the integral from 0 to infinite, the magnetic field is equals to equals nu not i by 4 pi r k -cap.

03:13 Interestingly this is half of the magnetic field due to infinite wire.

03:18 Two long segments have a magnetic field along positive z direction, therefore due to the ranked and rule.

03:25 And we have the magnetic field at point a, b, a equals nu0i by 4r, 1 plus 2 by pi as it is the addition of semicircle plus twice of 2 semi.

03:48 Now substituting the values into the equation 1 plus 2 by pi into 4 pi into 10 raised to the power minus 7 into current is 2 .2 ampers divided by 4 into r is 8 .5 and 1 that is 10 raise to the power minus 3 must be multiplied...

Please give Ace some feedback