University Physics II · topic UPII-09.03
Fields of loops, arcs, and coils
Connect the ideas, make your working visible, and test the result. Use the hints when you need a starting point; open the full solution after your attempt.
Further practice
Work through a fresh case before tackling the extension.
Fields of loops, arcs and coils: predict a finite change
The on-axis field of a circular coil is the Biot-Savart integral evaluated with the ring's symmetry: B = μ₀NIR²/2(R² + z²)^(3/2), pointing along the axis in the sense the right hand gives. Use the supplied topic model Y_out = 1885*r^2/(r^2+z^2)^1.5, where Y_out is FIELD AT THE PROBE (μT). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. z is Probe distance z on the axis, measured in cm. Keep r (Coil radius R) = 6 cm fixed. For this exercise use only 3.44375 cm ≤ z ≤ 21.5562 cm on this continuous branch. A grid of test settings is an input restriction for this exercise, not by itself a claim of physical quantisation. This calculation retains the topic model's stated assumptions; it does not extend them to a different physical setup. Calculate Y_out at z = 7.4285 cm and 9.96425 cm. Find the signed output change ΔY_out and the average sensitivity ΔY_out/Δz across this interval. State whether the response increases or decreases; distinguish this interval average from a derivative at one point.
FIELD AT THE PROBE against z
Predictions from the supplied Fields of loops, arcs and coils model, with all other inputs fixed as stated in the question. 5 calibration points support a trend comparison; they are not measured data or a straight-line law.
Vertical axis (y): FIELD AT THE PROBE / μT
Horizontal axis (x): z: Probe distance z on the axis / cm
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| z: Probe distance z on the axis / cm | FIELD AT THE PROBE / μT |
|---|---|
| 3.44375 | 204.9570159456949 |
| 7.971874999999999 | 68.31983085838122 |
| 12.499999999999998 | 25.45741677182713 |
| 17.028125 | 11.531168152754821 |
| 21.55625 | 6.057368723296098 |
These are the supplied values, not a worked answer. No fitted trend is drawn.
Show a hint
Substitute each value of z separately while keeping every other input fixed, then subtract in the same order.
Reveal the worked solution
- At z = 7.4285 cm, substitute into Y_out = 1885*r^2/(r^2+z^2)^1.5: Y₀ = 77.9374 μT.
- At z = 9.96425 cm, the same fixed-parameter model gives Y₁ = 43.1254 μT.
- Signed change: ΔY_out = Y₁ − Y₀ = -34.812 μT; the input change is Δz = 2.53575 cm.
- Average sensitivity is ΔY_out/Δz = -13.7285 μT per cm. Its negative sign means that this response decreases over the specified interval.
- This is a secant slope between two settings. It equals a point derivative only for a locally linear relation or in a justified small-interval limit.
Answer Y₀ = 77.9374 μT, Y₁ = 43.1254 μT; ΔY_out = -34.812 μT; average sensitivity = -13.7285 μT per cm.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.
Extra challenging questions
Work through the linked stages and explain what each result means.
Fields of loops, arcs and coils: infer an input with uncertainty
The on-axis field of a circular coil is the Biot-Savart integral evaluated with the ring's symmetry: B = μ₀NIR²/2(R² + z²)^(3/2), pointing along the axis in the sense the right hand gives. Use the supplied topic model Y_out = 1885*r^2/(r^2+z^2)^1.5, where Y_out is FIELD AT THE PROBE (μT). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. z is Probe distance z on the axis, measured in cm. Keep r (Coil radius R) = 6 cm fixed. For this exercise use only 3.44375 cm ≤ z ≤ 21.5562 cm on this continuous branch. A grid of test settings is an input restriction for this exercise, not by itself a claim of physical quantisation. This calculation retains the topic model's stated assumptions; it does not extend them to a different physical setup. A reading gives Y_out = 17.584684 μT, with a bounded reading uncertainty ±1.7291 μT. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown z by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.00181125 cm, using a numerical bracket if required. Find the full input interval consistent with the bounded reading uncertainty and compare it with a symmetric input-error assumption. Do not assume that the supplied calibration points prove monotonicity outside this branch.
FIELD AT THE PROBE against z
Predictions from the supplied Fields of loops, arcs and coils model, with all other inputs fixed as stated in the question. 5 calibration points support a trend comparison; they are not measured data or a straight-line law.
Vertical axis (y): FIELD AT THE PROBE / μT
Horizontal axis (x): z: Probe distance z on the axis / cm
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| z: Probe distance z on the axis / cm | FIELD AT THE PROBE / μT |
|---|---|
| 3.44375 | 204.9570159456949 |
| 7.971874999999999 | 68.31983085838122 |
| 12.499999999999998 | 25.45741677182713 |
| 17.028125 | 11.531168152754821 |
| 21.55625 | 6.057368723296098 |
These are the supplied values, not a worked answer. No fitted trend is drawn.
Show a hint
The model's endpoint predictions bracket the reading. Solve Y_out(z) = Y_measured and then solve again at Y_measured − u and Y_measured + u; a nonlinear inverse can give unequal input deviations.
Reveal the worked solution
- Endpoint predictions are Y_out(3.44375) = 204.957 μT and Y_out(21.5562) = 6.05737 μT. The target lies between them. On the selected branch the response is strictly decreasing.
- Endpoint interpolation gives z_line = 3.44375 + (17.5847 − 204.957)(21.5562 − 3.44375)/(6.05737 − 204.957) = 20.5065 cm. This estimates the inverse of the endpoint chord, rather than assuming that chord is the physical model.
- Use the actual equation, not the chord, and bracket its root. Evaluating the midpoint and retaining the sign-changing half-interval gives z = 14.4924 cm; substituting back gives Y_out = 17.584684 μT.
- The interpolation error in the input is |z_line − z| = 6.01416 cm. This is larger than the requested numerical tolerance.
- Invert both limits Y_out = 15.855584 μT and 19.313784 μT. Sort the roots even if the response decreases: 13.9684 cm ≤ z ≤ 15.0866 cm.
- The lower and upper input deviations are 0.523964 cm and 0.594224 cm. They differ because the inverse response is nonlinear; a symmetric error bar would only be a local approximation. This interval includes reading uncertainty only, not model error or uncertainty in fixed inputs.
Answer z ≈ 14.4924 cm; compatible interval [13.9684, 15.0866] cm. Endpoint interpolation gives 20.5065 cm.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.