Explanations 
Textbooks 
Exams 
GCSE 
IGCSE 
AS 
A Level 
International A Level 
University Admissions Tests 
Flashcards 
Vaia AI 
Notes 
Study Plans 
Study Sets 
Find a job 
Find a degree 
Student Deals 
Magazine 
Mobile App Chapter 7: Problem 19
Two flat coils each of 10 turns have mean radii of 20 and \(2 \mathrm{~cm}\). Find an approximate value for the force between them if they are coaxial, \(10 \mathrm{~cm}\) apart and if each carries \(5 \mathrm{~A}\) in the same sense.
Short Answer
Expert verified
Answer: The approximate value of the force between the two flat coils is 0.1 N.
Step by step solution
01
Find the magnetic field produced by one coil at the center of the other coil
To find the magnetic field produced by one coil at the center of the other coil, we can use Ampère's circuital law for a circular loop with a rectangular path.
Formula: \(B = \frac{\mu_0 NI}{2a}\), where \(N\) is the number of turns, \(I\) is the current, \(a\) is the radius of the coil, and \(\mu_0 \approx 4\pi \times 10^{-7} T\cdot m/A\) is the permeability of free space.
For the larger coil with a radius of \(20\mathrm{~cm}\) or \(0.2\mathrm{~m}\):
\(B = \frac{4\pi \times 10^{-7} \cdot 10 \cdot 5}{2 \cdot 0.2} = 5 \times 10^{-6} \mathrm{T}\)
02
Find the magnetic field produced by one coil at the location of the other coil
To find the magnetic field produced by one coil at the location of the other coil, we can scale the result obtained in step 1 by the ratio of the distance between the coils to the radius of the larger coil.
Scaled magnetic field: \(B' = B \cdot \frac{0.2}{0.1} = 5\times 10 ^{-6} \mathrm{T} \cdot \frac{0.2}{0.1} = 10 ^{-5} \mathrm{T}\)
03
Find the force between the two coils
We can now use the formula for the force between two coaxial coils to find the force between the coils.
Formula: \(F = \frac{\mu_0 NI^2 R_2^2}{4\pi d^2}\), where \(N\) is the number of turns, \(I\) is the current, \(R_2\) is the radius of the smaller coil, and \(d\) is the distance between the coils.
\(F = \frac{4\pi \times 10^{-7} \cdot 10 \cdot 5^2 \cdot (0.02)^2}{4\pi \cdot (0.1)^2} = \frac{10 \times 100 \times 10^{-6}}{0.01} = 0.1 \mathrm{N}\)
The approximate value of the force between the two flat coils is \(0.1 \mathrm{N}\).
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with Vaia!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Ampère's Circuital Law
Ampère's circuital law is a fundamental principle in electromagnetism that describes the relationship between the current flowing through a conductor and the magnetic field it generates. This law is essential when studying the interactions between electrical currents and magnetic fields, such as in the case of analyzing the force between two coils.
The law can be stated simply: The line integral of the magnetic field (\textbf{B}) around a closed loop is proportional to the electric current (I) passing through the loop. Mathematically, it is expressed as \[oint \textbf{B} \bullet d\textbf{l} = \frac{\text{{4}}\text{{\text{{\text{\text{{\text{\text{\text{\text{{\text{\text{{\text{\text{{\text{\text{{\text{\text{{\text{\text{{\text{\text{\text{\text{{\text{\text{\text{{\text{\text{{\text{\text{\text{{\text{\text{\text{{\text{\text{\text{{\text{\text{\text{{\text{\text{\text{{\text{\text{\text{{\text{\text{\text{{\text{\text{{\text{\text{\text{{\text{\text{\text{{\text{\text{\text{{\text{\text{\text{{\text{\text{\text{{\text{\text{\text{{\text{\text{\text{pi}}}\text{{...}}}\text{{\text{due to each of the five U-form fragments; the two being circumferential, and the remaining three radial or linear; the line treatment of these fragments--resulting. to obtain the standard formula...}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}} \](...)where \textbf{B} is the magnetic field vector, d\textbf{l} is the differential length vector around the loop, and \text{{4}}\text{{\text{{\text{\text{{\text{\text{\text{{\text{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{\text{{\text{\text{{\text{\text{{\text{\text{{\text{\text{{\text{\text{\text{{\text{\text{\text{{\text{\text{{\text{\text{{\text{\text{{\text{\text{{\text{\text{{\text{\text{{\text{\text{\text{{\text{\text{{\text{\text{{\text{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{...{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{...ющих для масс. высокую эффективность.
The law can be stated simply: The line integral of the magnetic field (\textbf{B}) around a closed loop is proportional to the electric current (I) passing through the loop. Mathematically, it is expressed as \[oint \textbf{B} \bullet d\textbf{l} = \frac{\text{{4}}\text{{\text{{\text{\text{{\text{\text{\text{\text{{\text{\text{{\text{\text{{\text{\text{{\text{\text{{\text{\text{{\text{\text{\text{\text{{\text{\text{\text{{\text{\text{{\text{\text{\text{{\text{\text{\text{{\text{\text{\text{{\text{\text{\text{{\text{\text{\text{{\text{\text{\text{{\text{\text{\text{{\text{\text{{\text{\text{\text{{\text{\text{\text{{\text{\text{\text{{\text{\text{\text{{\text{\text{\text{{\text{\text{\text{{\text{\text{\text{pi}}}\text{{...}}}\text{{\text{due to each of the five U-form fragments; the two being circumferential, and the remaining three radial or linear; the line treatment of these fragments--resulting. to obtain the standard formula...}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}} \](...)where \textbf{B} is the magnetic field vector, d\textbf{l} is the differential length vector around the loop, and \text{{4}}\text{{\text{{\text{\text{{\text{\text{\text{{\text{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{\text{{\text{\text{{\text{\text{{\text{\text{{\text{\text{{\text{\text{\text{{\text{\text{\text{{\text{\text{{\text{\text{{\text{\text{{\text{\text{{\text{\text{{\text{\text{{\text{\text{\text{{\text{\text{{\text{\text{{\text{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{...{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{\text{{...ющих для масс. высокую эффективность.
