University Physics II · unit UPII-02
Electric Fields
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.
Extra challenging questions
A selection from this pathway. Open a topic below for its complete practice and challenge set.
Electric field & the test charge: infer an input with uncertainty
The electric field at a point is the force a charge would feel there divided by that charge, taken in the limit of a vanishingly small positive test charge, so that the result describes the point and not the probe. Use the supplied topic model Y_out = 6+9.98617*q/k, where Y_out is SEPARATION (cm). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. k is Stiffness of the source mount, measured in mN m⁻¹. Keep q (Probe charge q₀) = 2 nC fixed. For this exercise use only 60.625 mN m⁻¹ ≤ k ≤ 139.375 mN m⁻¹ 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 = 6.1838016 cm, with a bounded reading uncertainty ±0.0060753 cm. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown k by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.007875 mN m⁻¹, 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.
SEPARATION against k
Predictions from the supplied Electric field & the test charge 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): SEPARATION / cm
Horizontal axis (x): k: Stiffness of the source mount / mN m⁻¹
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| k: Stiffness of the source mount / mN m⁻¹ | SEPARATION / cm |
|---|---|
| 60.625 | 6.329440659793814 |
| 80.3125 | 6.248682832684825 |
| 100 | 6.1997234 |
| 119.6875 | 6.166870725848564 |
| 139.375 | 6.14329930044843 |
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(k) = 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(60.625) = 6.32944 cm and Y_out(139.375) = 6.1433 cm. The target lies between them. On the selected branch the response is strictly decreasing.
- Endpoint interpolation gives k_line = 60.625 + (6.1838 − 6.32944)(139.375 − 60.625)/(6.1433 − 6.32944) = 122.24 mN m⁻¹. 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 k = 108.662 mN m⁻¹; substituting back gives Y_out = 6.1838016 cm.
- The interpolation error in the input is |k_line − k| = 13.5774 mN m⁻¹. This is larger than the requested numerical tolerance.
- Invert both limits Y_out = 6.1777263 cm and 6.1898769 cm. Sort the roots even if the response decreases: 105.186 mN m⁻¹ ≤ k ≤ 112.377 mN m⁻¹.
- The lower and upper input deviations are 3.47676 mN m⁻¹ and 3.71446 mN m⁻¹. 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 k ≈ 108.662 mN m⁻¹; compatible interval [105.186, 112.377] mN m⁻¹. Endpoint interpolation gives 122.24 mN m⁻¹.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.
Fields of Point Charges: infer an input with uncertainty
The electric field of a point charge is a vector defined at every point in space, directed along the line from the charge to that point, with magnitude k|q|/r² — outward if the charge is positive, inward if it is negative. Use the supplied topic model Y_out = 89.9*qmag/(r*r), where Y_out is FIELD E (kN C⁻¹). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. r is Probe distance r, measured in cm. Keep qmag (Charge magnitude |q|) = 2 nC; s (Sign of q (−1 or +1)) = 1 fixed. For this exercise use only 0.978125 cm ≤ r ≤ 4.52187 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 = 18.238212 kN C⁻¹, with a bounded reading uncertainty ±1.4167 kN C⁻¹. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown r by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.000354375 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 E against r
Predictions from the supplied Fields of Point Charges 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 E / kN C⁻¹
Horizontal axis (x): r: Probe distance r / cm
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| r: Probe distance r / cm | FIELD E / kN C⁻¹ |
|---|---|
| 0.9781249999999999 | 187.93210097071528 |
| 1.8640624999999997 | 51.74504250486038 |
| 2.75 | 23.77520661157025 |
| 3.6359375 | 13.600562445047759 |
| 4.521875 | 8.793314003330774 |
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(r) = 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(0.978125) = 187.932 kN C⁻¹ and Y_out(4.52187) = 8.79331 kN C⁻¹. The target lies between them. On the selected branch the response is strictly decreasing.
- Endpoint interpolation gives r_line = 0.978125 + (18.2382 − 187.932)(4.52187 − 0.978125)/(8.79331 − 187.932) = 4.33503 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 r = 3.13981 cm; substituting back gives Y_out = 18.238212 kN C⁻¹.
- The interpolation error in the input is |r_line − r| = 1.19522 cm. This is larger than the requested numerical tolerance.
- Invert both limits Y_out = 16.821512 kN C⁻¹ and 19.654912 kN C⁻¹. Sort the roots even if the response decreases: 3.02454 cm ≤ r ≤ 3.26936 cm.
- The lower and upper input deviations are 0.115273 cm and 0.129544 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 r ≈ 3.13981 cm; compatible interval [3.02454, 3.26936] cm. Endpoint interpolation gives 4.33503 cm.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.
Field superposition and null points: infer an input with uncertainty
Superposition says the electric field at a point is the vector sum of the fields each source charge would produce there on its own, and a null point is a place where that sum comes to zero. Use the supplied topic model Y_out = 8.99*(q1/(0.4/(1+s*sqrt(q2/q1)))+s*q2/abs((0.4/(1+s*sqrt(q2/q1)))-0.4)), where Y_out is POTENTIAL V THERE (V). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; sqrt(x) means √x; abs(x) means |x|. q2 is Size of q₂ at x = 0.40 m, measured in nC. Keep q1 (Charge q₁ at x = 0) = 8 nC; s (Sign of q₂: −1 opposite, +1 same) = -1 fixed. For this exercise use only 1.2125 nC ≤ q2 ≤ 2.7875 nC 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 = 41.217924 V, with a bounded reading uncertainty ±1.6554 V. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown q2 by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.0001575 nC, 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.
POTENTIAL V THERE against q2
Predictions from the supplied Field superposition and null points 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): POTENTIAL V THERE / V
Horizontal axis (x): q2: Size of q₂ at x = 0.40 m / nC
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| q2: Size of q₂ at x = 0.40 m / nC | POTENTIAL V THERE / V |
|---|---|
| 1.2125 | 67.05495809344706 |
| 1.6062499999999997 | 54.768668272875075 |
| 1.9999999999999998 | 44.95000000000001 |
| 2.39375 | 36.895126431500834 |
| 2.7874999999999996 | 30.182235722945645 |
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(q2) = 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(1.2125) = 67.055 V and Y_out(2.7875) = 30.1822 V. The target lies between them. On the selected branch the response is strictly decreasing.
- Endpoint interpolation gives q2_line = 1.2125 + (41.2179 − 67.055)(2.7875 − 1.2125)/(30.1822 − 67.055) = 2.31612 nC. 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 q2 = 2.17325 nC; substituting back gives Y_out = 41.217924 V.
- The interpolation error in the input is |q2_line − q2| = 0.142866 nC. This is larger than the requested numerical tolerance.
- Invert both limits Y_out = 39.562524 V and 42.873324 V. Sort the roots even if the response decreases: 2.09458 nC ≤ q2 ≤ 2.25501 nC.
- The lower and upper input deviations are 0.0786655 nC and 0.0817562 nC. 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 q2 ≈ 2.17325 nC; compatible interval [2.09458, 2.25501] nC. Endpoint interpolation gives 2.31612 nC.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.
Electric field lines: infer an input with uncertainty
An electric field line is a curve drawn so that the electric field is tangent to it at every point, arrowed in the direction of the force on a positive test charge. Use the supplied topic model Y_out = deg(atan2(py/hypot(px,py)^3 + q*py/hypot(px-8,py)^3, px/hypot(px,py)^3 + q*(px-8)/hypot(px-8,py)^3)), where Y_out is TANGENT ANGLE AT P (° from +x). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; hypot(a,b) means √(a²+b²); deg converts radians to degrees; atan2(y,x) is the angle of the vector (x,y), in radians. py is Probe P, height above the axis, measured in cm. Keep q (Charge on B, in units of q_A) = -1; px (Probe P, distance right of A) = 2 cm fixed. For this exercise use only 0.8975 cm ≤ py ≤ 3.1025 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 = 37.51514 ° from +x, with a bounded reading uncertainty ±0.53324 ° from +x. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown py by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.0002205 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.
TANGENT ANGLE AT P against py
Predictions from the supplied Electric field lines 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): TANGENT ANGLE AT P / ° from +x
Horizontal axis (x): py: Probe P, height above the axis / cm
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| py: Probe P, height above the axis / cm | TANGENT ANGLE AT P / ° from +x |
|---|---|
| 0.8975 | 20.53419956537583 |
| 1.44875 | 29.62716580032207 |
| 1.9999999999999998 | 35.67529651749391 |
| 2.5512499999999996 | 39.262782912852295 |
| 3.1024999999999996 | 41.07004898158468 |
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(py) = 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(0.8975) = 20.5342 ° from +x and Y_out(3.1025) = 41.07 ° from +x. The target lies between them. On the selected branch the response is strictly increasing.
- Endpoint interpolation gives py_line = 0.8975 + (37.5151 − 20.5342)(3.1025 − 0.8975)/(41.07 − 20.5342) = 2.7208 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 py = 2.24255 cm; substituting back gives Y_out = 37.51514 ° from +x.
- The interpolation error in the input is |py_line − py| = 0.478248 cm. This is larger than the requested numerical tolerance.
- Invert both limits Y_out = 36.9819 ° from +x and 38.04838 ° from +x. Sort the roots even if the response decreases: 2.16595 cm ≤ py ≤ 2.32591 cm.
- The lower and upper input deviations are 0.076601 cm and 0.0833617 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 py ≈ 2.24255 cm; compatible interval [2.16595, 2.32591] cm. Endpoint interpolation gives 2.7208 cm.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.