Magnetic induction of circular loop at axial point

BREAD · Mar 12, 2017

Homework Statement

Typed homework problem 6 about a circular wire loop of radius a in the XY-plane, centred at the origin and carrying current I, asking for the magnetic induction on the axis and for derivatives of B_y and components of B near the axis.

Homework Equations

The Attempt at a Solution

Handwritten solution to a magnetic field problem: a diagram of a current loop of radius R with axis z and point P at height z, followed by the Biot-Savart expressions for dB and the axial field Bz = mu0 I a^2 / (2 (a^2 + z^2)^(3/2)), and part (b) asking for the partial derivative of Bz with respect to y.

This is what i did about (a)
but i don't know how to approach (b), it doesn't have x,y components, only Bz component.

haruspex · Mar 13, 2017

I think your answer to a is right - not expert in this area.
For b, a dependent variable being zero at some value of the independent variable does not imply that its derivative is also zero there.
I suggest working on b2, returning to b1 afterwards.

BREAD · Mar 13, 2017

Handwritten derivation: by symmetry dBy/dy = dBx/dx; div B = 0 gives dBz/dz + dBy/dy + dBz/dz = 0, so dBy/dy = -1/2 dBz/dz; then dBy/dy = (3μ0I/4)(a^2 z)/(a^2 + z^2)^(5/2).

I think i solve (1), but the problem is (2),(3)

haruspex · Mar 13, 2017

BREAD said:

I think i solve (1)

And very neatly too.

BREAD said:

the problem is (2),(3)

These are easy. Fundamental differential calculus: f(x+Δx)≈f(x)+f'(x)Δx.

BREAD · Mar 13, 2017

haruspex said:

And very neatly too.

These are easy. Fundamental differential calculus: f(x+Δx)≈f(x)+f'(x)Δx.

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Handwritten derivation on lined paper: a Taylor expansion of B_z(0, Δy, z) − B_z(0, 0, z) ≈ ∂B_z/∂y at (0,0,z) times Δy, with ∂B_z/∂y and ∂B_y/∂z circled, and curl-zero relations ∂B_z/∂y = ∂B_y/∂z listed on the right.

I tried it , but is (2) right? and idw how to get ∂(B_z)/∂y|_(0, 0, z). I heard somewhere that by properties of curl of B (as i wrote)
∂(B_z)/∂y|_(0, 0, z) is changed to ∂(B_y)/∂z|_(0, 0, z), but i can't solve this.

haruspex · Mar 13, 2017

BREAD said:

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View attachment 114508

I tried it , but is (2) right? and idw how to get ∂(B_z)/∂y|_(0, 0, z). I heard somewhere that by properties of curl of B (as i wrote)
∂(B_z)/∂y|_(0, 0, z) is changed to ∂(B_y)/∂z|_(0, 0, z), but i can't solve this.

Isn't B_y identically zero on the z axis?

Magnetic induction of circular loop at axial point

Homework Statement

Homework Equations

The Attempt at a Solution

Discussion

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