Physics · Ch 3 — Magnetism and Magnetic Effects of Electric Current

Magnetic Field Due to a Long Current Carrying Solenoid

3.9.3

Magnetic Field Due to a Long Current Carrying Solenoid

To find the field inside a long solenoid of $N$ turns over length $L$ (turns per unit length $n=N/L$ ), choose a rectangular Amperian loop $abcd$ with side $ab$ (length $h$ , taken equal to the full solenoid length $L$ for convenience) lying inside the solenoid parallel to the axis, side $cd$ lying entirely outside, and sides $bc$ , $da$ crossing perpendicular to the field. Since $B=0$ outside, the integral along $cd$ is zero; since $bc$ and $da$ are perpendicular to $\vec B$ , those two integrals are also zero ( $\cos90°=0$ ); only the $ab$ leg survives, contributing $BL$ . Ampere's law with the enclosed current $NI$ (all $N$ turns pass through the loop) then gives $BL = \mu_0 NI$ , so

$\boxed{B = \mu_0\,\frac{N}{L}\,I = \mu_0\,n\,I}$ …

Figure 3.41Amperian loop for a solenoid

What this figure shows. A long solenoid is drawn in cross-section with its field lines pointing along its axis inside and spreading out negligibly outside. A rectangular Amperian loop abcd is superimposed so that side ab (length h) lies inside the solenoid parallel to the axis, side cd lies entirely outside, and sides bc and da cross from inside to outside, perpendicular to the field -- the specific loop shape used to isolate just …

Misc Example 3.16How B changes with length and turns of a solenoid

Worked out. Starting from B = mu0 N I / L for a solenoid of N turns over length L: (a) if only the length doubles (L to 2L) with N fixed, the new field is B' = mu0 N I/(2L) = B/2, i.e. it halves. (b) If both L and N double together, the turns-per-length ratio N/L is unchanged, so B' = mu0 (2N) I/(2L) = mu0 N I/L = B, i.e. it stays the same. (c) If N doubles but L stays fixed, B' = mu0(2N)I/L = 2B, i.e. it doubles. Comparing the three, B(L,2N) > B(L,N) > B(2L,N): packing more turns into the same length increases the field, stretching the same turns over a longer length decreases it, and sca …

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