Chapter 29: Q62P (page 861)

Question: In Figure, current $I = 56.2 mA$ is set up in a loop having two radial lengths and two semicircles of radii $a = 5.72 cm$ and $b = 9.36 cm$ with a common center $P$ (a) What are the magnitude and (b) What are the direction (into or out of the page) of the magnetic field at P and the (c) What is the magnitude of the loop’s magnetic dipole moment? and (d) What is the direction of the loop’s magnetic dipole moment?

Short Answer

Expert verified

Magnitude of magnetic field at point P is $4.97 \times 10^{- 7} T$ .
Direction of magnetic field at point P is into the page.
Magnitude of the loop’s magnetic dipole moment is $1.06 \times 10^{- 3} {Am}^{2}$ .
Direction of dipole moment is into the page.

Step by step solution

Identification of given data

Current is $56.2 mA$
Radius $a = 5.72 cm$
Radius is $b = 9.36 cm$

Understanding the concept of magnetic field for semicircle

Semicircle wire generates a magnetic field that is half that of circle wire. We use the formula for magnetic field for semicircle, and the direction is given by right hand rule.

Formula:

$% M a t h T y p e! T r a n s l a t o r! 2! 1! A M S L a T e X . t d l! A M S L a T e X! % M a t h T y p e! M T E F! 2! 1! + - % f e a a g K a r t 1 e v 2 a q a t C v A U f e B S j u y Z L 2 y d 9 g z L b v y N v 2 C a e r b b j x A H X % g a r m W u 51 M y V X g a r u a v P 1 w z Z b I t L D h i s 9 w B H 5 g a r u W q V v N C P v M C G 4 u z % 3 b q e e 0 e v G u e E 0 j x y a i b a i e Y l f 9 i r V e e u 0 d X d h 9 v q q j - h E e e u 0 x X d b b % a 9 f r F j 0 - O q F f e a 0 d X d d 9 v q a q - J f r V k F H e 9 p g e a 0 d X d a r - J b 9 h s 0 d X d % b P Y x e 9 v r 0 - v r 0 - v q p W q a a e a a b i G a a i a a c a W a b e a a e a W a a u a a a O q a a a % b a a a a a a a a a p e G a a m O q a i a b g 2 d a 9 m a a l a a a p a q a a 8 q a c q a H 8 o q B c a W G % j b G a e q i U d e h a p a q a a 8 q a c a a I 0 a G a e q i W d a N a a m O u a a a a a a a! 41 C 1! $ B = \ f r a c {{\ m u I \ t h e t a}} {{4 \ p i R}} $ % M a t h T y p e! E n d! 2! 1!$ uncaught exception: Invalid chunk

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(a) Determining the Magnitude of the magnetic field at point P

Now magnetic field for semicircle is as follows

$B_{1} = \frac{μ_{o} I θ}{4 πa} = \frac{μ_{o} I π}{4 πa} = \frac{μ_{o} I}{4 a}$

And for radius b it is as follows

$B_{2} = \frac{μ_{o} I}{4 b}$

So total magnetic field is as follows

role="math" localid="1662799565933" $B = B_{1} + B_{2} = \frac{μ_{o} I}{4 a} + \frac{μ_{o} I}{4 b} = \frac{μ_{o} I}{4} (\frac{1}{a} + \frac{1}{b}) = (4 π \times 10^{- 7} {NA}^{- 2}) \times (56.2 \times 10^{- 3} A) (\frac{1}{0.0572 m} + \frac{1}{0.0936 m}) = 4.97 \times 10^{- 7} T$