The Biot-Savart law is a fundamental principle in magnetostatics, which is a branch of physics that deals with electromagnetic force. It is an equation that describes the magnetic field generated by a constant electric current. In other words, it relates the magnetic field to the magnitude, direction, length, and proximity of the electric current. This law is applicable in the magnetostatic estimate and is consistent with both Gauss's law for magnetism and Ampere's circuital law. The law was discovered by French physicists Jean-Baptiste Biot and Félix Savart in 1820.
To apply the Biot-Savart law, one must consider the magnetic field generated by a current-carrying conductor, such as a wire or a coil. The law states that the magnetic induction at a point near a current-carrying conductor is directly proportional to the current in the element (I), the length of the element (dl), the sine of the angle between the element and the line joining the element to the point, and inversely proportional to the square of the distance between the element and the point. By using this law, one can calculate the magnetic field produced by a current element, as well as the magnetic field due to various configurations such as a circular coil, a disk, or a line segment.
Characteristics | Values |
---|---|
What it is | A mathematical expression that illustrates the magnetic field produced by a stable electric current in electromagnetism |
What it does | Relates the magnetic field to the magnitude, length, direction, and proximity of the electric current |
Application 1 | Used to calculate magnetic responses at the atomic or molecular level |
Application 2 | Used in aerodynamic theory to calculate the velocity induced by vortex lines |
Importance 1 | Similar to Coulomb's law in electrostatics |
Importance 2 | Applicable for very small conductors that carry current |
Importance 3 | Applicable for symmetrical current distribution |
What You'll Learn
- Calculating the magnetic field of a circular arc of wire
- Magnetic field strength at a point produced by a current-carrying wire
- Magnetic field produced by a constant electric current
- Magnetic field in an indefinitely long straight wire due to a constant current
- Magnetic field in the centre of a current-carrying arc
Calculating the magnetic field of a circular arc of wire
The Biot-Savart law is an equation that gives the magnetic field produced by a current-carrying segment. This law enables us to calculate the magnitude and direction of the magnetic field produced by a current in a wire.
To calculate the magnetic field of a circular arc of wire, we can use the Biot-Savart law. Consider a circular arc of wire with radius R carrying a current I. The magnetic field at the centre of this arc at point P can be calculated as follows:
Firstly, we apply the Biot-Savart law equation:
\[\stackrel{\to }{\textbf{B}}=\frac{{\mu }_{0}}{4\pi }\underset{\text{wire}}{\int }\frac{Id\stackrel{\to }{\textbf{l}}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\hat{\textbf{r}}}{{r}^{2}}..\]
Since the contributions to the magnetic field are in the same direction (out of the page), we can work with the magnitude of the field. The cross product turns into multiplication because the path dl and the radial direction are perpendicular. We can also substitute the arc length formula, dl=rdθ:
\[B=\frac{{\mu }_{0}}{4\pi }\underset{\text{wire}}{\int }\frac{Ir\phantom{\rule{0.2em}{0ex}}d\theta }{{r}^{2}}..\]
The current and radius can be taken out of the integral because they are the same regardless of where we are on the path. This leaves only the integral over the angle:
\[B=\frac{{\mu }_{0}I}{4\pi r}\underset{\text{wire}}{\int }d\theta .\]
The angle varies on the wire from 0 to θ; hence, the result is:
\[B=\frac{{\mu }_{0}I\theta }{4\pi r}.\]
The direction of the magnetic field at point P is determined by the right-hand rule. If there are other wires in the diagram, find each contribution from a wire or arc and add the results by superposition of vectors. Note that in a symmetric situation, like a circular wire, contributions from opposite sides of point P cancel each other.
This calculation assumes that the medium is not magnetic (e.g. a vacuum). The Biot-Savart law is valid in the magnetostatic approximation and is consistent with Ampère's circuital law and Gauss's law for magnetism.
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Magnetic field strength at a point produced by a current-carrying wire
The Biot-Savart law is an equation that describes the magnetic field generated by a constant electric current. It is fundamental to magnetostatics and is valid in the magnetostatic approximation. The law is named after Jean-Baptiste Biot and Félix Savart, who discovered the relationship in 1820.
The Biot-Savart law is used to compute the resultant magnetic flux density at a position in 3D space generated by a filamentary current (e.g. due to a wire). The equation in SI units is:
\\displaystyle \\mathbf {B} (\\mathbf {r} )={\\frac {\\mu _{0}}{4\\pi }}\\int _{C}{\\frac {I\\,d{\\boldsymbol {\\ell }}\\times \\mathbf {r'} }{|\\mathbf {r'} |^{3}}}}
Where:
- \\displaystyle \\mathbf {B} (\\mathbf {r} ) is the magnetic field vector at position r
- \\displaystyle {\\frac {\\mu _{0}}{4\\pi }} is a constant, with \\displaystyle \\mu _{0} being the magnetic constant
- \\displaystyle \\int _{C} denotes the line integral over the path C of the electric current
- \\displaystyle I is the current
- \\displaystyle d{\\boldsymbol {\\ell }} is a vector along the path C whose magnitude is the length of the differential element of the wire in the direction of the conventional current
- \\displaystyle {\\boldsymbol {\\ell }} is a point on path C
- \\displaystyle \\mathbf {r'} =\\mathbf {r} -{\\boldsymbol {\\ell }} is the full displacement vector from the wire element at point \\displaystyle {\\boldsymbol {\\ell }} to the point at which the field is being computed
The Biot-Savart law can be applied to calculate the magnetic field at a specific point in space due to a current-carrying wire. The steps to do this are:
- Choose the point in space where the magnetic field is to be calculated.
- Holding that point fixed, calculate the line integral over the path of the electric current to find the total magnetic field at that point.
- Determine the direction of the magnetic field using the right-hand rule, where the thumb points in the direction of the conventional current and the other fingers show the direction of the magnetic field.
For example, consider a short wire of length 1.0 cm carrying a current of 2.0 A in the vertical direction. To calculate the magnetic field at a point P that is 1 meter from the wire in the x-direction, the Biot-Savart law can be used as follows:
\\displaystyle B=\\frac{{\\mu }_{0}}{4\\pi }\\phantom{\\rule{0.2em}{0ex}}\\frac{I\\text{Δ}l\\phantom{\\rule{0.2em}{0ex}}\\text{sin}\\phantom{\\rule{0.1em}{0ex}}\\theta }{{r}^{2}}=\\left(1\\phantom{\\rule{0.2em}{0ex}}×\\phantom{\\rule{0.2em}{0ex}}{10^{-7}\\text{T}\\.cdot \\text{m/A}}\\right)\\left(\\frac{2\\phantom{\\rule{0.2em}{0ex}}\\text{A}\\left(0.01\\phantom{\\rule{0.2em}{0ex}}\\text{m}\\right)\\phantom{\\rule{0.1em}{0ex}}\\text{sin}\\left(89.4\\text{°}\\right){{\\left(1\\phantom{\\rule{0.2em}{0ex}}\\text{m}}\\right)^{2}}\\right)=2.0\\phantom{\\rule{0.2em}{0ex}}×\\phantom{\\rule{0.2em}{0ex}}{10^{-9}}\\text{T}
Where:
- \\displaystyle B is the magnetic field at point P
- \\displaystyle {\\mu }_{0}=1\\phantom{\\rule{0.2em}{0ex}}×\\phantom{\\rule{0.2em}{0ex}}{10^{-7}\\text{T}\\.cdot \\text{m/A} is the permeability of free space
- \\displaystyle I=2\\text{ A} is the current
- \\displaystyle \\text{Δ}l=0.01\\text{ m} is the length of the wire
- \\displaystyle r=1\\text{ m} is the distance from the wire to point P
- \\displaystyle \\theta =89.4\\text{°} is the angle between \\displaystyle \\text{Δ}l and the vector from the wire to point P
The Biot-Savart law can also be used to calculate the magnetic field produced by more complex geometries, such as a current in a circular arc. For example, consider a wire carrying a current I in a circular arc with radius R swept through an arbitrary angle \\displaystyle \\theta. The magnetic field at the centre of this arc at point P can be calculated as follows:
\\displaystyle B=\\frac{{\\mu }_{0}I\\theta }{4\\pi r}
Where:
- \\displaystyle B is the magnetic field at point P
- \\displaystyle {\\mu }_{0} is the permeability of free space
- \\displaystyle I is the current
- \\displaystyle \\theta is the angle of the circular arc
- \\displaystyle r is the radius of the circular arc
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Magnetic field produced by a constant electric current
The Biot-Savart law is an equation that describes the magnetic field generated by a constant electric current. It relates the magnetic field to the magnitude, direction, length, and proximity of the electric current.
The law is fundamental to magnetostatics and is valid in the magnetostatic approximation. It is also consistent with Ampère's circuital law and Gauss's law for magnetism. The Biot-Savart law is expressed as:
$$\left| dB\right|=(\frac{\mu _0}{4\pi })(\frac{Ids\,sin\theta }{r^2})$$
Where:
- Μ0 is the permeability of free space
- I is the current
- Ds is a small element of the wire length
- R is the position vector of the point in question drawn from the current element
- Θ is the angle between r and ds
The direction of the magnetic field is always in a plane perpendicular to the line of the element and position vector. This can be determined using the right-hand thumb rule, where the thumb points in the direction of the conventional current, and the other fingers show the direction of the magnetic field.
The Biot-Savart law can be used to calculate the magnetic field produced by a constant electric current in various configurations, such as a long straight wire, a circular loop, or a solenoid.
For example, consider a long straight current-carrying wire. The magnetic field strength produced by such a wire can be calculated using the formula:
$$B = \frac{{\mu _0}I}{{2\pi r}}$$
Where:
- B is the magnetic field strength
- R is the distance from the wire
The direction of the magnetic field around a long straight wire can be determined using the right-hand rule, where the thumb points in the direction of the current, and the curled fingers indicate the direction of the magnetic field loops.
Now, let's consider a circular loop of wire carrying a current. The magnetic field strength at the centre of the loop can be calculated using the formula:
$$B = \frac{{\mu _0}I}{{2R}}$$
Where:
- B is the magnetic field strength
- R is the radius of the loop
The direction of the magnetic field near the loop can be determined using the right-hand rule, and its complexity can be further explored using compasses or Hall probes.
A solenoid is a long coil of wire with many turns or loops. The magnetic field inside a solenoid is remarkably uniform in magnitude and direction, while the field just outside the coils is nearly zero. The magnetic field strength inside a solenoid is given by:
$$B = {\mu _0}nI$$
Where:
- B is the magnetic field strength
- N is the number of loops per unit length of the solenoid
- I is the current
The Biot-Savart law provides a fundamental understanding of the magnetic fields generated by constant electric currents and serves as a basis for further exploration and applications in the field of electromagnetism.
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Magnetic field in an indefinitely long straight wire due to a constant current
The Biot-Savart law is an equation that gives the magnetic field produced by a current-carrying conductor. It is fundamental to magnetostatics and is valid in the magnetostatic approximation. The law is named after Jean-Baptiste Biot and Félix Savart, who discovered this relationship in 1820.
The Biot-Savart law can be used to determine the magnetic field produced by a current in a straight wire. A moving charge, or current, produces a magnetic field. The magnetic field produced is composed of closed concentric circles around the wire. The strength of the resulting magnetic field can be determined using the following equation:
\[B = \dfrac{\mu_0 I}{2\pi R}\]
Where:
- \(B\) is the strength of the magnetic field
- \(\mu_0\) is the permeability of free space (\(4\pi \times 10^{-7}\) T·m/A)
- \(I\) is the current
- \(R\) is the shortest distance to the wire
It is important to note that the distance must be measured perpendicular to the wire. The strength of the magnetic field decreases as the distance from the wire increases. This is because the distance appears in the denominator of the equation, so the two values are inversely proportional.
The direction of the magnetic field produced by a long straight wire can be determined using the right-hand rule. Point the thumb of the right hand in the direction of the current, and the fingers will curl in the direction of the magnetic field loops.
To calculate the magnetic field produced by a straight wire, we can follow these steps:
- Identify the given values: current (\(I\)) and distance from the wire (\(R\)).
- Plug the given values into the equation: \(B = \dfrac{\mu_0 I}{2\pi R}\).
- Calculate the numerical value of \(\mu_0\) by multiplying the given value (\(4\pi \times 10^{-7}\) T·m/A) with the unit of current (ampere).
- Substitute the value of \(\mu_0 I\) into the equation.
- Simplify the equation by cancelling out any units that appear in both the numerator and denominator.
- Evaluate the resulting equation to find the strength of the magnetic field (\(B\)).
By applying the Biot-Savart law and using the right-hand rule, we can determine both the magnitude and direction of the magnetic field produced by a current in an indefinitely long straight wire.
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Magnetic field in the centre of a current-carrying arc
The Biot-Savart law is an equation that gives the magnetic field produced by a current-carrying segment. It is fundamental to magnetostatics and relates the magnetic field to the magnitude, direction, length, and proximity of the electric current.
The formula for the magnetic field produced at a point due to a small element of the wire is:
|dB| = (μ0 / 4π) * (Idsinθ / r^2)
Where:
- Μ0 is the permeability of free space
- I is the current
- Ds is a small element of the wire
- R is the position vector of the point in question drawn from the current element
- Θ is the angle between r and ds
The direction of the magnetic field is given by the right-hand thumb rule, where the thumb points in the direction of the conventional current, and the other fingers show the direction of the magnetic field.
Now, let's apply this to find the magnetic field in the centre of a current-carrying arc. We are given an arc of wire carrying the current through it, and we want to find the magnetic field at the centre, which is a distance 'r' from the arc. The Biot-Savart law states that the magnetic field at 'r' is proportional to the current flowing through the element, the length of the element, and the sine of the angle between the element and 'r', and inversely proportional to the square of 'r'.
Mathematically, this can be represented as:
B = (μ0 * I * l * sin(φ)) / (4π * r^2)
Where:
- B is the magnetic field at the centre of the arc
- Μ0 is the permeability of the medium in a vacuum
- I is the current carried by the arc
- L is the length of the arc
- Φ is the angle between the point and the current
Let's assume the angle between the point and the current is 90 degrees, so sin(φ) = 1. The length of the arc can be calculated as l = R * θ, where R is the radius of the arc, and θ is the angle the arc curves through.
Substituting these values into the formula, we get:
B = (μ0 * I * R * θ) / (4π * R^2)
Simplifying this expression, we find:
B = (μ0 * I) / (4R)
Therefore, the magnetic field in the centre of a current-carrying arc is given by:
B = (μ0 * I) / (4R)
This result demonstrates how the Biot-Savart law can be applied to calculate the magnetic field produced by a current-carrying arc at a specific point, in this case, the centre of the arc.
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