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Ampere's Law: Magnetic Field Due to a Long Straight Wire

Current running through a wire will produce a magnetic field that can be calculated using the Biot-Savart Law.

Learning Objective

  • Express the relationship between the strength of a magnetic field and a current running through a wire in a form of equation


Key Points

    • Ampere's Law states that for a closed curve of length C, magnetic field (B) is related to current (IC): $\oint_C {Bd\ell = \mu _0 I_C }$. In this equation, dl represents the differential of length of wire in the curved wire, and μ0 is the permeability of free space.
    • Ampere's Law can be related to the Biot-Savart law, which holds for a short, straight length of conductor: $d {\bf B}=\frac {\mu_0}{4 \pi} \frac {Id{\bf l} \times {\bf r}}{r^3}$. In this equation, partial magnetic field (dB) is expressed as a function of current for an infinitesimally small segment of wire (dl) at a point r distance away from the conductor.
    • After integrating, the direction of the magnetic field according to the Biot-Savart Law can be determined using the right hand rule.

Terms

  • magnetic field

    A condition in the space around a magnet or electric current in which there is a detectable magnetic force, and where two magnetic poles are present.

  • electric field

    A region of space around a charged particle, or between two voltages; it exerts a force on charged objects in its vicinity.


Full Text

Current running through a wire will produce both an electric field and a magnetic field. For a closed curve of length C, magnetic field (B) is related to current (IC) as in Ampere's Law, stated mathematically as:

$\oint_C {Bd\ell = \mu _0 I_C }$

In this equation, dl represents the differential of length of wire in the curved wire, and μ0 is the permeability of free space. This can be related to the Biot-Savart law. For a short, straight length of conductor (typically a wire) this law generally calculates partial magnetic field (dB) as a function of current for an infinitesimally small segment of wire (dl) at a point r distance away from the conductor:

$d {\bf B}=\frac {\mu_0}{4 \pi} \frac {Id{\bf l} \times {\bf r}}{r^3}$.

In this equation, the r vector can be written as r̂ (the unit vector in direction of r), if the r3 term in the denominator is reduced to r2 (this is simply reducing like terms in a fraction). Integrating the previous differential equation, we find:

${\bf B}=\frac {\mu_0}{4 \pi} \oint_C {\frac {Id{\bf l} \times {\bf \hat{r}}}{r^2}}$.

This relationship holds for constant current in a straight wire, in which magnetic field at a point due to all current elements comprising the straight wire is the same. As illustrated in the direction of the magnetic field can be determined using the right hand rule—pointing one's thumb in the direction of current, the curl of one's fingers indicates the direction of the magnetic field around the straight wire.

Direction of magnetic field

The direction of the magnetic field can be determined by the right hand rule.

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