University Physics II · topic UPII-09.08
Magnetic-field superposition and mapping
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.
Magnetic-Field Superposition & Mapping: predict a finite change
Magnetic-field superposition is the rule that the field at a point equals the vector sum of the fields each current would produce there by itself, with every other current switched off. Use the supplied topic model Y_out = 157.08*cur*(1/(1+(zp/12-s/24)^2)^1.5+1/(1+(zp/12+s/24)^2)^1.5), where Y_out is PROBE READS B(z) (μT). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. zp is Probe position z, measured in cm. Keep s (Coil separation s) = 12 cm; cur (Current in each coil) = 2 A fixed. For this exercise use only -11.8125 cm ≤ zp ≤ -0.7875 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 zp = -9.387 cm and -7.8435 cm. Find the signed output change ΔY_out and the average sensitivity ΔY_out/Δzp across this interval. State whether the response increases or decreases; distinguish this interval average from a derivative at one point.
PROBE READS B(z) against zp
Predictions from the supplied Magnetic-Field Superposition & Mapping 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): PROBE READS B(z) / μT
Horizontal axis (x): zp: Probe position z / cm
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| zp: Probe position z / cm | PROBE READS B(z) / μT |
|---|---|
| -11.8125 | 283.80331658853544 |
| -9.05625 | 361.96000261367413 |
| -6.3 | 420.8503663712551 |
| -3.5437499999999993 | 446.0151561452325 |
| -0.7874999999999996 | 449.57963329722975 |
These are the supplied values, not a worked answer. No fitted trend is drawn.
Show a hint
Substitute each value of zp separately while keeping every other input fixed, then subtract in the same order.
Reveal the worked solution
- At zp = -9.387 cm, substitute into Y_out = 157.08*cur*(1/(1+(zp/12-s/24)^2)^1.5+1/(1+(zp/12+s/24)^2)^1.5): Y₀ = 353.105 μT.
- At zp = -7.8435 cm, the same fixed-parameter model gives Y₁ = 391.641 μT.
- Signed change: ΔY_out = Y₁ − Y₀ = 38.5365 μT; the input change is Δzp = 1.5435 cm.
- Average sensitivity is ΔY_out/Δzp = 24.9669 μT per cm. Its positive sign means that this response increases 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 = 353.105 μT, Y₁ = 391.641 μT; ΔY_out = 38.5365 μT; average sensitivity = 24.9669 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.
Magnetic-Field Superposition & Mapping: infer an input with uncertainty
Magnetic-field superposition is the rule that the field at a point equals the vector sum of the fields each current would produce there by itself, with every other current switched off. Use the supplied topic model Y_out = 157.08*cur*(1/(1+(zp/12-s/24)^2)^1.5+1/(1+(zp/12+s/24)^2)^1.5), where Y_out is PROBE READS B(z) (μT). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. zp is Probe position z, measured in cm. Keep s (Coil separation s) = 12 cm; cur (Current in each coil) = 2 A fixed. For this exercise use only -11.8125 cm ≤ zp ≤ -0.7875 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 = 435.93495 μT, with a bounded reading uncertainty ±2.0467 μT. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown zp by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.0011025 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.
PROBE READS B(z) against zp
Predictions from the supplied Magnetic-Field Superposition & Mapping 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): PROBE READS B(z) / μT
Horizontal axis (x): zp: Probe position z / cm
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| zp: Probe position z / cm | PROBE READS B(z) / μT |
|---|---|
| -11.8125 | 283.80331658853544 |
| -9.05625 | 361.96000261367413 |
| -6.3 | 420.8503663712551 |
| -3.5437499999999993 | 446.0151561452325 |
| -0.7874999999999996 | 449.57963329722975 |
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(zp) = 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(-11.8125) = 283.803 μT and Y_out(-0.7875) = 449.58 μT. The target lies between them. On the selected branch the response is strictly increasing.
- Endpoint interpolation gives zp_line = -11.8125 + (435.935 − 283.803)(-0.7875 − -11.8125)/(449.58 − 283.803) = -1.69494 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 zp = -5.08725 cm; substituting back gives Y_out = 435.93495 μT.
- The interpolation error in the input is |zp_line − zp| = 3.39231 cm. This is larger than the requested numerical tolerance.
- Invert both limits Y_out = 433.88825 μT and 437.98165 μT. Sort the roots even if the response decreases: -5.29032 cm ≤ zp ≤ -4.8631 cm.
- The lower and upper input deviations are 0.20307 cm and 0.224146 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 zp ≈ -5.08725 cm; compatible interval [-5.29032, -4.8631] cm. Endpoint interpolation gives -1.69494 cm.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.