Magnetic Fields due to Conductors, Loops, and Solenoids - Tenth Grade Science
In 1820, the Danish physicist Hans Christian Oersted discovered that an electric current flowing through a wire deflects a nearby magnetic compass needle. This breakthrough demonstrated that electricity and magnetism are interconnected phenomena. By shaping conducting wires into straight lines, circular coils, or cylindrical solenoids, engineers can control magnetic field patterns, providing the basis for electromagnets, electric motors, relays, and magnetic resonance imaging.
Magnetic Field due to a Straight Current-Carrying Conductor
When a steady electric current flows through a straight metallic wire, it generates a magnetic field in the surrounding space consisting of concentric circular lines centered on the wire.
CONCENTRIC FIELD AROUND A STRAIGHT WIRE
Current (I) ^
|
.-----.|
.' <-- |'.
/ | \
| * | | Magnetic Field Lines
\ | | / (Concentric Circles)
'. v .|.'
'-----'|
|
Geometric Characteristics
The magnetic field lines lie in planes perpendicular to the straight conducting wire. The centers of all circular field lines reside along the axis of the wire. As the distance ($r$) from the wire increases, the concentric circles become progressively wider and more spaced out, indicating that magnetic field strength decreases with distance.
The Right-Hand Thumb Rule (Maxwell's Corkscrew Rule)
The directional relationship between current and the resulting magnetic field is determined by the Right-Hand Thumb Rule:
Imagine grasping the straight current-carrying conductor with your right hand such that your outstretched thumb points in the direction of conventional current. Your curled fingers will naturally encircle the conductor in the exact direction of the magnetic field lines.
RIGHT-HAND THUMB RULE
Thumb points WITH Current (I)
^
|
.---|---.
/ .-'-. \
| | | | Curled fingers indicate
| | | | direction of Circular
\ '-.-' / Magnetic Field (B)
'---|---'
|
Factors Determining Field Strength
The magnitude of the magnetic field ($B$) at a radial distance $r$ from a long straight conductor carrying current $I$ obeys the relation:
$$B \propto \frac{I}{r} \implies B = \frac{\mu_0 I}{2 \pi r}$$
Here, $\mu_0$ is the magnetic permeability of free space ($\mu_0 = 4\pi \times 10^{-7} \text{ T}\cdot\text{m/A}$). The field strength increases proportionally with current ($I$) and decreases inversely with radial distance ($r$).
Magnetic Field due to a Circular Current-Carrying Loop
When a straight wire is bent into a circular loop, every segment of the loop generates its own circular magnetic field lines.
MAGNETIC FIELD OF A CIRCULAR CURRENT LOOP
Current UP
^
|
.----+----.
.' | '.
/ (B) | \
| <-----*-----> | Field lines are nearly
| | | STRAIGHT and PARALLEL
\ | (B) / at the center of loop
'. | .'
'----+----'
|
v
Current DOWN
Superposition at the Loop Center
Near the wire perimeter, the field lines form small concentric circles. As you move toward the center of the circular loop, the arcs of these circles become larger and flatter. At the exact geometric center of the loop, the contributions from all opposing wire segments reinforce each other constructively, resulting in magnetic field lines that are straight, parallel, and perpendicular to the plane of the loop.
Field Strength of an $N$-Turn Coil
If a circular coil contains $N$ closely wound turns of wire, each turn carries the same current in the same direction. Because magnetic fields add vectorially, the total magnetic field at the center is $N$ times stronger than that of a single turn:
$$B_{\text{center}} = N \times \frac{\mu_0 I}{2 R}$$
Here, $R$ is the radius of the circular coil, and $N$ is the number of turns.
The Clock Face Rule
To determine the magnetic polarity of a circular face of a current loop, use the Clock Face Rule: If looking at the face of the coil reveals current circulating in an anticlockwise direction, that face behaves as a magnetic North Pole ($N$). If looking at the face reveals current circulating in a clockwise direction, that face behaves as a magnetic South Pole ($S$).
THE CLOCK FACE RULE
NORTH POLE SOUTH POLE
(Anticlockwise Current) (Clockwise Current)
^ --->
.' '. .' '.
/ \ / \
| N | | S |
\ / \ /
'. .' '. .'
v <---
Magnetic Field due to a Solenoid
A solenoid is a long helical coil of insulated copper wire wound closely into a cylindrical shape.
MAGNETIC FIELD PATTERN OF A SOLENOID
S N
+-------------------------------------------------+
<------| ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) |<------
(South) +-------------------------------------------------+ (North)
| |
+--------------[/ ]-------------( + - )-----------+
Switch Battery
Inside the Solenoid: Field is UNIFORM (Parallel straight lines)
Outside: Identical to the magnetic field of a BAR MAGNET
Field Geometry Inside and Outside
The magnetic field pattern produced by a current-carrying solenoid is nearly identical to that of a permanent bar magnet: Outside the solenoid, field lines emerge from the north-polar face, loop through external space, and enter the south-polar face. Inside the core volume of the solenoid, the field lines are straight, parallel, and densely packed along the central axis. This indicates that the internal magnetic field is uniform in magnitude and direction throughout the core.
Internal Magnetic Field Formula
The magnetic field strength ($B$) inside an ideal, tightly wound long solenoid is given by:
$$B = \mu_0 n I = \mu_0 \left(\frac{N}{L}\right) I$$
Here, $n = N / L$ represents the turn density (number of turns per unit length), $I$ is the current, and $\mu_0$ is the permeability of free space. The field strength can be increased by increasing the turn density or raising the current.
Electromagnets
An electromagnet is a temporary magnet created by inserting a ferromagnetic core (typically soft iron) into the interior cavity of a current-carrying solenoid.
ANATOMY OF AN ELECTROMAGNET
Soft Iron Core (High Mu)
=====================
Insulated Copper --( ( ( ( ( ( ( ( ( ( ( )-- Current Leads
Coil Wire =====================
Soft Iron vs. Steel Cores
Soft iron is chosen as the core material for electromagnets because it has high magnetic permeability and low magnetic retentivity. When current flows, the soft iron core magnetizes quickly, multiplying the solenoid's magnetic field by several hundred times. When the current is switched off, the soft iron loses almost all of its induced magnetization immediately.
In contrast, hardened steel possesses high retentivity and high coercivity. Once magnetized by a solenoid, it retains its magnetism even after the current is removed, making steel suitable for permanent magnets rather than temporary electromagnets.
ELECTROMAGNET VS PERMANENT MAGNET
Property Electromagnet Permanent Magnet
----------------------------------------------------------------------
Magnetism Temporary (Switchable) Permanent (Persistent)
Field Strength Adjustable (via I and n) Fixed once manufactured
Polarity Reversible (Reverse I) Fixed North and South
Core Material Soft Iron Hardened Carbon Steel, Alnico
Chapter Exercises and Solutions
Exercise 1: A horizontal electrical power line suspended 4.0 meters above the ground carries a steady direct current of 100 amperes flowing from east to west. Using the right-hand thumb rule, determine the direction of the resulting magnetic field at a point directly beneath the transmission wire on the ground.
Solution 1: Position your right hand such that your outstretched thumb points horizontally toward the west, matching the direction of conventional current. As your fingers curl naturally around the horizontal conductor, they point downward on the northern side of the wire, sweep horizontally toward the south underneath the wire, and point upward on the southern side. Therefore, at a geographic point located directly beneath the transmission line on the ground, the curled fingers point toward the south. The magnetic field at that ground position is directed horizontally from north to south.
Exercise 2: An insulated copper wire solenoid of length 0.50 meters contains 1000 closely wound turns and carries an electric current of 2.0 amperes. Calculate the turn density of the solenoid and determine the internal magnetic field strength, given that $\mu_0 = 4\pi \times 10^{-7} \text{ T}\cdot\text{m/A}$.
Solution 2: The turn density $n$ represents the number of turns per unit length: $n = N / L = 1000 \text{ turns} / 0.50 \text{ m} = 2000 \text{ turns/meter}$. The internal magnetic field of an air-core solenoid is given by $B = \mu_0 n I$. Substituting the parameters gives $B = (4\pi \times 10^{-7} \text{ T}\cdot\text{m/A}) \times (2000 \text{ m}^{-1}) \times (2.0 \text{ A}) = 1.6\pi \times 10^{-3} \text{ T} \approx 5.03 \times 10^{-3} \text{ T} = 5.03 \text{ mT}$. The uniform magnetic field within the core of the solenoid has an intensity of approximately 5.03 millitesla.
Exercise 3: Explain why a soft iron cylinder is inserted inside the coil of an electric scrap crane, and explain what operational problem would arise if hardened steel were used instead.
Solution 3: An electric scrap crane requires a powerful magnetic field that can be turned on to lift heavy scrap iron and turned off to drop the load into a transport bin. A soft iron core has high magnetic permeability, which amplifies the solenoid's magnetic field to lift heavy loads. Soft iron also has low magnetic retentivity, meaning that as soon as the operator turns off the current, the core demagnetizes immediately, allowing the scrap metal to drop. If hardened steel were used instead, its high retentivity would cause it to become permanently magnetized. After turning off the electric current, the steel core would continue to attract the scrap metal, making it impossible to release the load.