Deriving expression for magnetic field at point P due to current

• jisbon
In summary, the formula for the magnetic field at a distance x from a wire carrying current I, with the wire at a distance a from the origin and oriented along the y-axis, is given by B(x) = (mu_0*I)/(4*pi*x)*(a/(3*sqrt(x^2+(a^2/9))) - (2*a)/(3*sqrt(x^2+(4*a^2/9)))). This can be simplified for large values of x to give an inverse square law. The correctness of the formula was confirmed by checking for a typo.
jisbon
Homework Statement
Derive an expression for the magnetic field at point P due to current-carrying wire of length a
Relevant Equations
NIL

So I think I do understand how to do this, but the thing is my answers are always incorrect. Will need some guidance/help on this.
##B =\frac{\mu_{0}I}{4\pi}\int\frac{dysin\theta}{r^2} ##
##y = Rtan\phi##
##dy = Rsec^2\phi d\phi##
##B = \frac{\mu_{0}I}{4\pi}\int\frac{dysin\theta}{r^2} = \frac{\mu_{0}I}{4\pi}\int\frac{Rsec^2\phi d\phi sin(\frac{\pi}{2}-\phi)}{R^2sec^2 \phi}
= \frac{\mu_{0}I}{4\pi}\int\frac{cos\phi}{R}d\phi = \frac{\mu_{0}I}{4\pi}(sin\phi_{2}-sin\phi_{1}) ##

So,
##sin\phi_{1}=\frac{2/3a}{\sqrt{4/9a^2+x^2}}##
##sin\phi_{2}=\frac{1/3a}{\sqrt{1/9a^2+x^2}} ##
##\frac{\mu_{0}I}{4\pi}(sin\phi_{2}-sin\phi_{1}) = \frac{\mu_{0}I}{4\pi} (\frac{1/3a}{\sqrt{1/9a^2+x^2}} - \frac{2/3a}{\sqrt{4/9a^2+x^2}})##

Is this correct?

Delta2
I think it is correct, just a typo you forgot the term ##\frac{1}{x}## or ##\frac{1}{R}## so the correct formula should be
$$B(x)=\frac{\mu_0 I}{4\pi x}(\frac{a}{3\sqrt{x^2+\frac{a^2}{9}}}-\frac{2a}{3\sqrt{x^2+\frac{4a^2}{9}}})$$One rule of thumb to check such expressions (for static electric or magnetic fields) is that for distances far away from the source (that must have finite dimensions) , you should get approximately an inverse square law, and such is the case here, for ##x## large (in comparison with ##a##) the ##x^2## term dominates in the square roots so each square root approximately simplifies to ##\sqrt{(x^2+0)}=x## and together with the other ##x## from outside the parenthesis we get an inverse square law.

Last edited:

1. What is the formula for calculating the magnetic field at a point due to current?

The formula for calculating the magnetic field at a point due to current is: B = (μ0/4π) * (I * dl * sinθ / r^2), where μ0 is the permeability of free space, I is the current, dl is the length of the current-carrying wire, θ is the angle between the wire and the line connecting the wire to the point, and r is the distance between the wire and the point.

2. How do you derive the expression for magnetic field at point P due to current?

To derive the expression for magnetic field at point P due to current, we use the Biot-Savart Law, which states that the magnetic field at a point is directly proportional to the current and the length of the current-carrying wire, and inversely proportional to the distance from the wire to the point. We also use the Right-Hand Rule to determine the direction of the magnetic field.

3. What is the significance of the angle θ in the formula for magnetic field?

The angle θ in the formula for magnetic field represents the angle between the current-carrying wire and the line connecting the wire to the point where the magnetic field is being measured. This angle affects the strength of the magnetic field, as the magnetic field is strongest when the wire is perpendicular to the line connecting it to the point, and weakest when the wire is parallel to the line.

4. How does the distance from the current-carrying wire to the point affect the magnetic field?

The distance from the current-carrying wire to the point affects the magnetic field as it is inversely proportional to the square of the distance. This means that the magnetic field decreases as the distance from the wire to the point increases. This is why the magnetic field is strongest when the point is close to the wire, and weakest when the point is far from the wire.

5. What is the unit of measurement for the magnetic field at a point due to current?

The unit of measurement for the magnetic field at a point due to current is Tesla (T). However, in some cases, the unit Gauss (G) is also used, with 1 T = 10,000 G. This unit represents the strength of the magnetic field at a specific point in space.

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