Sources of Magnetic Fields

The Biot-Savart law gives the magnetic field contributed by a short piece of a current: its size falls as the inverse square of the distance and as sinθ, and its direction is the cross product of the element with the direction to the field point. Use the supplied topic model Y_out = 2000*cur*sind(th)/(d*d), where Y_out is dB AT P (nT). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; sind/cosd/tand use degrees. d is Distance r to the field point, measured in cm. Keep cur (Current I) = 10 A; th (Angle θ from the element) = 60 ° fixed. For this exercise use only 3.31875 cm ≤ d ≤ 5.68125 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 = 764.48599 nT, with a bounded reading uncertainty ±34.179 nT. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown d by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.00023625 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.

The field of a long straight wire is azimuthal: at every point outside the conductor it is tangent to the circle through that point centred on the wire, with magnitude μ₀I/2πr, where r is the perpendicular distance from the axis. Use the supplied topic model Y_out = 20*i/d, 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. d is Probe distance d, measured in cm. Keep i (Current I) = 12 A fixed. For this exercise use only 3.6375 cm ≤ d ≤ 8.3625 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 = 36.811227 µT, with a bounded reading uncertainty ±1.2168 µT. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown d by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.0004725 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.

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.

A solenoid is a wire wound in a close helix, whose interior magnetic field is axial and uniform with magnitude B = μ₀nI, set by the turns per metre n and the current I and by nothing else. Use the supplied topic model Y_out = len/(2*sqrt(len^2+r^2)), where Y_out is AT THE MOUTH (dimensionless). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; sqrt(x) means √x. len is Winding length ℓ, measured in cm. Keep r (Winding radius R) = 2 cm; z (Probe position z) = 0 cm fixed. For this exercise use only 6.7625 cm ≤ len ≤ 27.2375 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 = 0.49732367, with a bounded reading uncertainty ±0.00020007. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown len by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.0020475 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.