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12.2 Magnetic Field Due to a Thin Straight Wire

2 min read•june 24, 2024

Magnetic fields around wires are a key concept in electromagnetism. The Biot-Savart law helps us calculate these fields, showing how they depend on current and distance. Understanding this relationship is crucial for many electrical applications.

The is a handy tool for visualizing magnetic fields around wires. It shows how the field forms around the wire, with strength decreasing as you move farther away. This knowledge is essential for working with electromagnets and motors.

Magnetic Field of a Thin Straight Wire

Biot-Savart law for straight wires

Top images from around the web for Biot-Savart law for straight wires

  • 12.3 Magnetic Force between Two Parallel Currents – University Physics Volume 2 View original

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  • 22.9 Magnetic Fields Produced by Currents: Ampere’s Law – College Physics: OpenStax View original

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Top images from around the web for Biot-Savart law for straight wires

  • 12.3 Magnetic Force between Two Parallel Currents – University Physics Volume 2 View original

    Is this image relevant?

  • 22.9 Magnetic Fields Produced by Currents: Ampere’s Law – College Physics: OpenStax View original

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  • 12.3 Magnetic Force between Two Parallel Currents – University Physics Volume 2 View original

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  • Calculates B⃗\vec{B} at a point in space due to using dB⃗=μ04πIdl⃗×r^r2d\vec{B} = \frac{\mu_0}{4\pi} \frac{I d\vec{l} \times \hat{r}}{r^2}
    • μ0\mu_0: (4π×10−7 T⋅m/A4\pi \times 10^{-7} \text{ T} \cdot \text{m/A})
    • II: current flowing through wire (amperes)
    • dl⃗d\vec{l}: infinitesimal length of wire (meters)
    • r^\hat{r}: unit vector pointing from wire segment to point in space
    • rr: distance from wire segment to point in space (meters)
  • Integrate Biot-Savart law over entire wire length for total magnetic field
    • : B=μ0I2πrB = \frac{\mu_0 I}{2\pi r}
    • of length LL: B=μ0I4πr(sin⁡θ2−sin⁡θ1)B = \frac{\mu_0 I}{4\pi r} (\sin \theta_2 - \sin \theta_1), θ1\theta_1 and θ2\theta_2 are angles between point and wire ends (radians)
  • provides an alternative method for calculating magnetic fields around current-carrying wires

Current and distance in field strength

  • Magnetic field strength directly proportional to current II through wire
    • Doubling current doubles magnetic field strength
  • Magnetic field strength inversely proportional to distance rr from wire
    • Doubling distance reduces magnetic field strength by factor of 2
    • Follows inverse square law for short wire segments, inverse linear relationship for infinite wires (falls off more slowly with distance)

Right-hand rule for wire fields

  • Determines direction of magnetic field around current-carrying wire
    • Thumb points in current flow direction
    • Fingers curl around wire in magnetic field direction
  • form concentric circles around wire
    • Circles lie in planes perpendicular to wire
    • Strongest field closest to wire, weakens with increasing distance
  • Reversing current direction reverses magnetic field direction
  • : process by which a changing magnetic field induces an electric current in a nearby conductor
  • Magnetic flux: measure of the total magnetic field passing through a given area
  • : a coil of wire that produces a uniform magnetic field when current flows through it
  • Magnetic dipole moment: a measure of the strength and orientation of a magnetic dipole (e.g., a current loop)
  • : imaginary lines used to visualize the direction and strength of a magnetic field
  • : the driving force for magnetic flux in a magnetic circuit

Key Terms to Review (30)

Ampere: An ampere (A) is the unit of electric current in the International System of Units (SI). It is defined as the flow of one coulomb of charge per second.
Ampere: The ampere (symbol: A) is the base unit of electric current in the International System of Units (SI). It is defined as the constant flow of one coulomb of electric charge per second, and it is a fundamental quantity in the study of electromagnetism and electrical circuits.
Ampère's Law: Ampère's law is a fundamental principle in electromagnetism that relates the magnetic field generated by an electric current to the magnitude and direction of that current. It is one of the four Maxwell's equations, which together describe the relationships between electric and magnetic fields and electric charges and currents.
B = μ₀I / (2πr): The equation B = μ₀I / (2πr) describes the relationship between the magnetic field (B) created by a thin, straight wire and the current (I) flowing through the wire, the permeability of free space (μ₀), and the distance (r) from the wire. This equation is fundamental in understanding the magnetic field due to a thin straight wire, as described in the physics topic 12.2 Magnetic Field Due to a Thin Straight Wire.
Circular Field Lines: Circular field lines refer to the pattern of magnetic field lines that surround a thin, straight wire carrying an electric current. These field lines form concentric circles around the wire, with the strength of the magnetic field decreasing as the distance from the wire increases.
Concentric Circles: Concentric circles are a set of circular shapes that share a common center point and have different radii, creating a series of nested circles. This geometric concept is particularly relevant in the context of understanding the magnetic field due to a thin straight wire.
Current-Carrying Wire: A current-carrying wire is a conductor that allows the flow of electric current, which in turn generates a magnetic field around the wire. This concept is central to understanding the Biot-Savart Law and the magnetic field due to a thin straight wire.
D𝐵⃗: d𝐵⃗ represents an infinitesimal change in the magnetic field vector, $\vec{B}$. It is a fundamental concept in the study of magnetic fields, particularly in the context of understanding the magnetic field due to a thin, straight wire as described in the topic 12.2 Magnetic Field Due to a Thin Straight Wire.
D𝑙⃗: The term d𝑙⃗ represents an infinitesimal vector element of length along a current-carrying wire, which is crucial for calculating the magnetic field produced by that wire. This vector points in the direction of the current and has a magnitude equal to an infinitesimally small segment of the wire. Understanding d𝑙⃗ allows for the application of the Biot-Savart law, which relates this small segment to the contribution it makes to the overall magnetic field at a point in space.
Electromagnetic Induction: Electromagnetic induction is the process by which a changing magnetic field induces an electromotive force (EMF) in a conductor, causing an electric current to flow. This phenomenon is fundamental to the operation of many electrical devices and is crucial in understanding the relationship between electricity and magnetism.
Finite Straight Wire: A finite straight wire is a segment of a straight wire with a finite length, as opposed to an infinitely long wire. It is a fundamental concept in understanding the magnetic field generated by current-carrying conductors.
Infinite straight wire: An infinite straight wire is an idealized concept in physics used to simplify calculations of electric and magnetic fields. It is assumed to extend infinitely in both directions with a uniform linear charge density.
Infinite Straight Wire: An infinite straight wire is a theoretical model used to represent a long, straight electrical conductor that extends infinitely in both directions. This idealized concept is central to understanding the magnetic fields generated by current-carrying wires, as described in the topics of the Biot-Savart Law and the Magnetic Field Due to a Thin Straight Wire.
Infinite Wire Approximation: The infinite wire approximation is a concept in electromagnetism where a long, straight wire is treated as an infinitely long conductor. This simplifies the calculation of the magnetic field generated by the wire, allowing for a more straightforward analysis of the magnetic field's properties and behavior.
Magnetic Field: A magnetic field is a region of space where magnetic forces can be detected. It is a fundamental concept in electromagnetism, describing the invisible lines of force that surround and permeate magnetic materials, electric currents, and changing electric fields. The magnetic field plays a crucial role in various topics within the study of college physics.
Magnetic field lines: Magnetic field lines are imaginary lines that represent the direction and strength of a magnetic field. They emerge from the north pole of a magnet and enter the south pole, forming continuous loops.
Magnetic Field Lines: Magnetic field lines are the invisible lines that represent the direction and strength of a magnetic field. They are used to visualize and understand the behavior of magnetic fields, which are crucial in various topics related to electromagnetism and electromagnetic induction.
Magnetic field vector: A magnetic field vector is a physical quantity that represents the strength and direction of a magnetic field at a specific point in space. It is depicted as an arrow, where the length indicates the strength of the field and the arrowhead shows the direction in which a north magnetic pole would move. This concept is essential for understanding how magnetic fields interact with electric currents, particularly when analyzing the magnetic field created by a thin straight wire carrying an electric current.
Magnetomotive Force: Magnetomotive force is a measure of the magnetic field strength created by an electric current or a permanent magnet. It is the driving force that produces a magnetic field and is a fundamental concept in understanding the behavior of magnetic circuits.
Permeability of free space: Permeability of free space, denoted as $\mu_0$, is a physical constant that describes the extent to which a magnetic field can penetrate and affect a vacuum. Its value is $4\pi \times 10^{-7}$ Tm/A.
Permeability of Free Space: The permeability of free space, denoted as $\mu_0$, is a fundamental physical constant that describes the magnetic properties of a vacuum or free space. It is a measure of the ability of free space to support the formation of a magnetic field in response to an electric current or changing electric field.
R̂: The unit vector in the direction of the position vector $\vec{r}$ from the current element to the point where the magnetic field is being calculated. It is a dimensionless vector that points in the same direction as the position vector $\vec{r}$ and has a magnitude of 1.
Right-hand rule: The right-hand rule is a mnemonic used to determine the direction of the magnetic field surrounding a current-carrying conductor. Point your thumb in the direction of the current and curl your fingers; your fingers indicate the direction of the magnetic field lines.
Right-Hand Rule: The right-hand rule is a mnemonic device used to determine the direction of various quantities related to electromagnetism, such as the direction of magnetic fields, the motion of charged particles in magnetic fields, and the direction of the magnetic force on a current-carrying conductor. It provides a simple and intuitive way to visualize and remember these directional relationships.
Solenoid: A solenoid is a coil of wire designed to create a uniform magnetic field in its interior when an electric current passes through it. It is commonly used in electromagnets, inductors, and valves.
Solenoid: A solenoid is a tightly wound coil of wire, often cylindrical in shape, that produces a magnetic field when an electric current passes through it. Solenoids are fundamental components in the study of electromagnetism and have applications in various areas of physics, including magnetic fields, magnetic force, and electromagnetic induction.
Tesla: The tesla (T) is the SI unit of magnetic flux density, representing the strength of a magnetic field. One tesla is defined as one weber per square meter.
Tesla: The tesla (T) is the unit of magnetic flux density, or magnetic field strength, in the International System of Units (SI). It is named after the Serbian-American inventor Nikola Tesla, who made significant contributions to the field of electromagnetism. The tesla is a fundamental unit that is essential in understanding and describing various electromagnetic phenomena and their applications.
Thin Straight Wire: A thin straight wire is a type of conductor that has a small diameter and extends in a straight line. It is an important concept in the study of magnetic fields and their interactions with electric currents.
μ₀: μ₀, also known as the permeability of free space or the vacuum permeability, is a fundamental physical constant that represents the magnetic permeability of free space or a vacuum. It is a crucial parameter in the study of electromagnetism and is used in various electromagnetic laws and equations.
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12.3 Magnetic Force between Two Parallel Currents