Biot-Savart law in electromagnetism

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Learn about the Biot-Savart Law in electromagnetism

Biot-Savart Law in Electromagnetism

The Biot-Savart Law is a fundamental principle in electromagnetism that relates the magnetic field generated by a current to the current distribution producing it. This law is named after the French mathematician Jean-Baptiste Biot and the French physicist Félix Savart, who independently discovered it in 1820.

Mathematical Formulation of the Law

The Biot-Savart Law is mathematically expressed as:

$$mathbf{B}(mathbf{r})=frac{mu_0}{4pi}int_C frac{I,dboldsymbol{ell}times(mathbf{r}-mathbf{r}^prime)}{|mathbf{r}-mathbf{r}^prime|^3}$$

where:

  • $$mathbf{B}(mathbf{r})$$ is the magnetic field at position $$mathbf{r}$$,
  • $$mu_0$$ is the permeability of free space,
  • $$I$$ is the current,
  • $$C$$ is the path of the current,
  • $$dboldsymbol{ell}$$ is an infinitesimal vector element along the path $$C$$,
  • $$mathbf{r}^prime$$ is the position of the current element, and
  • $$|mathbf{r}-mathbf{r}^prime|$$ is the distance between $$mathbf{r}$$ and

    Biot-Savart Law in Electromagnetism

    The Biot-Savart Law is a fundamental principle in electromagnetism that relates the magnetic field generated by a current to the current distribution producing it. This law is named after the French mathematician Jean-Baptiste Biot and the French physicist Félix Savart, who independently discovered it in 1820.

    Mathematical Formulation of the Law

    The Biot-Savart Law is mathematically expressed as:

    $$mathbf{B}(mathbf{r})=frac{mu_0}{4pi}int_C frac{I,dboldsymbol{ell}times(mathbf{r}-mathbf{r}^prime)}{|mathbf{r}-mathbf{r}^prime|^3}$$

    where:

    • $$mathbf{B}(mathbf{r})$$ is the magnetic field at position $$mathbf{r}$$,
    • $$mu_0$$ is the permeability of free space,
    • $$I$$ is the current,
    • $$C$$ is the path of the current,
    • $$dboldsymbol{ell}$$ is an infinitesimal vector element along the path $$C$$,
    • $$mathbf{r}^prime$$ is the position of the current element, and
    • $$|mathbf{r}-mathbf{r}^prime|$$ is the distance between $$mathbf{r}$$ and $$mathbf{r}^prime$$.

    This law is a vector equation, where the magnetic field $$mathbf{B}(mathbf{r})$$ is a vector quantity and is given by the vector product of $$dboldsymbol{ell}$$ and $$(mathbf{r}-mathbf{r}^prime)/|mathbf{r}-mathbf{r}^prime|^3$$.

    Applications of the Law

    The Biot-Savart Law has various applications in electromagnetism, including:

    Magnetic Field of a Straight Current-Carrying Wire

    The Biot-Savart Law can be used to calculate the magnetic field produced by a straight current-carrying wire. For a wire carrying a current $$I$$ and positioned along the $$z$$-axis, the magnetic field at a point $$P(x,y,z)$$ in the $$x-y$$ plane is given by:

    $$mathbf{B}(mathbf{r})=frac{mu_0I}{2pi}frac{hat{mathbf{phi}}}{r}$$

    where:

    • $$hat{mathbf{phi}}$$ is the azimuthal unit vector, and
    • $$r=sqrt{x^2+y^2}$$.

    Magnetic Field of a Circular Loop

    The Biot-Savart Law can also be used to calculate the magnetic field produced by a circular loop of wire. For a loop carrying a current $$I$$ and positioned in the $$x-y$$ plane, the