Ampere's Law in Computational Modeling of Electromagnetic Systems

Computational Modeling of Electromagnetic Systems

Class Overview

Computational Modeling of Electromagnetic Systems (F1014B) Ampere's Law 21st May 2025 Computational Modeling of Electromagnetic SystemsWhat is this class about?

• Understand the principles and mathematical form of Ampere's Law.
• Learn how to apply Ampere's Law to calculate magnetic fields in symmetric systems.
• Explore practical applications such as long straight wires, solenoids, and toroids.

21st May 2025 Computational Modeling of Electromagnetic Systems

Recap

Fundamental Concepts

2Recap 1. Homework 3. Fundamental Concepts

• Activity 2 Saturday 24th May B = NOI 4π dīxî +2 B= Xpolenc μo II'L F = 2πη

Magnetic Force, Field & Ampere's Law

2. Magnetic Force, Field & Ampere's Law

• Analyze the magnetic force on a straight or curved current-carrying conductor.
• Understand how moving charges generate magnetic fields, using the Biot-Savart Law.
• Apply Ampère's Law to find magnetic fields in systems with symmetry (e.g., wires, solenoids).

21st May 2025 Computational Modeling of Electromagnetic Systems 3Ampere's Law 21st May 2025 Computational Modeling of Electromagnetic Systems 4Ampere's Law $ B . di = polenc Bdlcos(0) = polenc B(2Tr) = polenc B = MOI 2πη B dl dl B B r I dl dl B 21st May 2025 Computational Modeling of Electromagnetic Systems 5Ampere's Law & B . di = polenc B dlcos(0) = polenc B(2Tr) = - polenc μoIenc B = - 2π B dl B B r I 1 dl di B 21st May 2025 Computational Modeling of Electromagnetic Systems 6Ampere's Law $ B . d = polenc ¿? b B BO B dl dl C dl > 12 B ni d> a I 21st May 2025 Computational Modeling of Electromagnetic Systems

Ampere's Law Principles

Perspective View and Current Direction

7Ampere's Law Perspective view 12 Arbitrary closed curve around conductors B Curl the fingers of your right hand around the integration path: Your thumb points in the direction of positive current. Top view Plane of curve 128 lenci = 11- 12+ 13 13 di B Ampere's law: If we calculate the line integral of the magnetic field around a closed curve, the result equals pro times the total enclosed current: 6B . dl = po lenci- 21st May 2025 Computational Modeling of Electromagnetic Systems

Ampere's Law Application

Cylindrical Conductor Example

8Ampere's Law Application Example

• A cylindrical conductor with radius R carries a current I. The current is uniformly distributed over the cross-sectional area of the conductor. Find the magnetic field as a function of the distance r from the conductor axis for points both inside (rR). B 2TR B = 2TT R2 B = 1 μο 1 2Tr 2 2TR 0 R 2R 3R 4R

21st May 2025 Computational Modeling of Electromagnetic Systems

Coaxial Cable Problem

9Ampere's Law Application Try by yourself

• Figure is a cross-sectional view of a coaxial cable. The center conductor is surrounded by a rubber layer, an outer conductor, and another rubber layer. In a particular application, the current in the inner conductor is I=1 A out of the page and the current in the outer conductor is 1=3 A into the page. Assuming the distance d=1mm, determine the magnitude and direction of the magnetic field at (a) point a and (b) point b. Claim a signature for this exercise. Three signatures = two points on the midterm. × × × 12 a × × 1 I 1 1 I I X 1 I * -- -- -- T d d d B = 200uT(top) B = 133pT(bottom)

21st May 2025 Computational Modeling of Electromagnetic Systems

Solenoid

10 × 1Solenoid B=0 C d Integration path L X X × X × × × × X × 8 B a Central part of solenoid $ B . di = polenc BL = polenc BL = NuoI B= XpoI 21st May 2025 Computational Modeling of Electromagnetic Systems

Toroidal Solenoid

11Toroidal Solenoid (a) (b) B X 1 X O Path 1 1 1 Path 2 Path 3 The magnetic field is confined almost entirely to the space enclosed by the windings (in blue). § B · dĺ = Molenc B(2Tr) = polenc B(2Tr) = NuoI B = ΝμΟΙ 2πr 21st May 2025 Computational Modeling of Electromagnetic Systems

Ampere's Law Application

Solenoid Current Calculation

12Ampere's Law Application Try by yourself

• A long solenoid that has 1000 turns uniformly distributed over a length of 0.400 m produces a magnetic field of magnitude 1.00x10-4T at its center. What current is required in the windings for that to occur? B=0 d 1. Integration path 1700 B b Y Central part of solenoid B= ¥poI Claim a signature for this exercise. Three signatures = two points on the midterm. I =31.8mA

21st May 2025 Computational Modeling of Electromagnetic Systems

Parallel Conductors Example

13Ampere's Law Application Example

• Four long, parallel conductors carry equal currents of I=5.00A. Figure is an end view of the conductors. The current direction is into the page at points A and B and out of the page at points C and D. Calculate (a) the magnitude and (b) the direction of the magnetic field at point P, located at the center of the square of edge length , l=0.2m. A × C . Il I P B × I I I D l B = 2 × 10-5T(bottom)

21st May 2025 Computational Modeling of Electromagnetic Systems

Webassign

14Webassign 21st May 2025 Computational Modeling of Electromagnetic Systems

Webassign 01 - Magnetic Field & Magnetic Force

15Webassign 01 - Magnetic Field & Magnetic Force : May 20 | 100 pts # Week 2 ...

Biot-Savart Law and Activity

00 03 - Biot-Savart Law ... Activity 2 - Biot Savart May 24 | 100 pts O ...

Webassign 02 - Sources of Magnetic Field

Webassign 02 - Sources of Magnetic Field 0 : May 27 | 100 pts 21st May 2025 Computational Modeling of Electromagnetic Systems

Midterm

16Midterm 21st May 2025 Computational Modeling of Electromagnetic Systems

Midterm Topics and Date

17

• Next Wednesday 28th May
• Topics:
• Magnetic Field & Magnetic Force (Lorentz Force)
• Sources of magnetic Fields (moving charges, currents)
• Biot-Savart Law
• Ampere's Law Applications
• Formulas will be provided by the teacher

21st May 2025 Computational Modeling of Electromagnetic Systems

Summary

Homework and Activities

18Summary 1. Homework 3. Fundamental Concepts

• Activity 2 Saturday 24th May Solenoid
• Webassign Activity Tuesday 27th May
• Midterm Wednesday 28th May B = N I MOI B = ΝμΟΙ 27Tr Toroidal Solenoid

Ampere's Law Summary

2. Ampere's Law

• The line integral of the magnetic field around a closed loop is proportional to the total current enclosed.
• Simplifies magnetic field calculations for geometries like straight wires, solenoids, and toroids.

21st May 2025 Computational Modeling of Electromagnetic Systems 19