📚 Potentiometer Circuits | 电位器电路
A potentiometer is an instrument that allows precise measurement of potential difference and electromotive force (e.m.f.) without drawing current from the source under test. In A-Level Physics, understanding potentiometer circuits is essential for comparing voltages, calibrating instruments, and determining internal resistance of cells. This article covers the operating principle, practical setups, calibration methods, and key precautions associated with the potentiometer.
电位计是一种能够在不从被测电源吸取电流的情况下精确测量电势差和电动势的仪器。在A-Level物理中，理解电位计电路对于比较电压、校准仪器以及测定电池内阻至关重要。本文涵盖电位计的工作原理、实际电路搭建、校准方法以及主要的注意事项。
1. What is a Potentiometer? | 什么是电位计？
A potentiometer, in the context of measurement, is not the same as a variable resistor used to control current. It consists of a long uniform resistance wire (usually 1 metre or more) stretched along a scale, a sliding contact (jockey), and a driver cell that maintains a steady current through the wire. The voltage drop along the wire is proportional to length, enabling the potentiometer to act as a precision adjustable voltage source.
在测量领域中，电位计不同于用来控制电流的可变电阻器。它由一根长的均匀电阻丝（通常1米或更长）、一个滑动触头（游标）、以及一个在电阻丝中维持稳定电流的驱动电池组成。沿着电阻丝的电压降与长度成正比，这使得电位计能够作为一个精密可调的电压源。
2. Potentiometer as a Voltage Divider | 电位计作为分压器
The resistance wire AB is connected at its ends to a driver cell E0 and a rheostat Rh in series. A steady current I flows through the wire, creating a potential drop VAB = I RAB. Any point J along the wire has a potential difference VAJ relative to A, which is a fraction of VAB directly proportional to the length AJ. By moving the jockey, we obtain any voltage between 0 and VAB. This is the voltage divider action.
电阻丝AB两端与驱动电池E0和一个变阻器Rh串联。稳定电流I流过电阻丝，产生的电势降为VAB = I RAB。电阻丝上任意一点J相对于A端的电势差VAJ是VAB的一部分，且与长度AJ成正比。移动游标，我们可以得到0到VAB之间的任何电压，这就是分压器的作用。
3. Principle of Operation – Null Deflection Method | 工作原理 – 零偏转法
The potentiometer measures an unknown e.m.f. εx by balancing it against the known potential drop along the wire. A galvanometer G is connected between the positive terminal of the cell under test and the jockey. When the jockey touches a point J such that VAJ exactly equals εx, no current flows through the galvanometer, and it shows zero deflection. At this balance point, the cell delivers no current, so its internal resistance has no effect and the measured value is the true e.m.f.
电位计通过将未知电动势εx与电阻丝上已知的电势降进行平衡来测量。电流计G连接在被测电池的正极与游标之间。当游标恰好接触到点J，使得VAJ精确等于εx时，电流计中无电流流过，指针为零偏转。在这个平衡点，电池不输出电流，因此其内阻没有影响，测量值就是真实的电动势。
4. The Potential Gradient along the Wire | 沿导线的电势梯度
The potential gradient k is defined as the voltage drop per unit length of the wire: k = VAB / L, where L is the total length AB. Its unit is V m-1 or V cm-1. A smaller potential gradient spreads the voltage over a longer length, making it easier to locate the balance point precisely. The gradient can be adjusted by changing the driver cell's output or the series rheostat, but it must remain constant during a set of measurements.
电势梯度k定义为单位长度上的电压降：k = VAB / L，其中L是AB的总长度。单位为V m-1或V cm-1。较小的电势梯度将电压分散在更长的距离上，从而更容易精确地找到平衡点。可以通过改变驱动电池的输出或串联变阻器来调整梯度，但在一组测量中必须保持恒定。
5. Measuring an Unknown EMF | 测量未知电动势
To measure εx, the potentiometer must first be calibrated. A standard cell of known e.m.f. εs (often a Weston cell with εs V) is connected in place of the unknown. The jockey is moved to find the balance length ls. The potential gradient is then k = εs / ls. Without changing the driver current, the unknown cell is substituted, and its balance length lx is found. Thus, the unknown e.m.f. is given by:
εx = εs × (lx / ls)
要测量εx，必须先校准电位计。将已知电动势的标准电池εs（通常是韦斯顿电池，εs = 1.0186 V）接入被测位置，移动游标找到平衡长度ls，则电势梯度k = εs / ls。不改变驱动电流，换上待测电池，找到其平衡长度lx。因此，未知电动势按下式计算：
εx = εs × (lx / ls)
6. Comparing Two EMFs | 比较两个电动势
If we only need the ratio of two e.m.f.s, absolute calibration is not necessary. The two cells ε1 and ε2 are connected through a selector switch to the galvanometer circuit. Balance lengths l1 and l2 are measured without altering the driving current. Since the same potential gradient applies, the ratio is simply ε1 / ε2 = l1 / l2. This method is useful for checking cells or determining if a cell's e.m.f. has dropped.
如果只需要两个电动势的比值，则不需要进行绝对校准。将两个电池ε1和ε2通过选择开关接入电流计回路，在不改变驱动电流的情况下分别测量平衡长度l1和l2。由于使用相同的电势梯度，比值即为ε1 / ε2 = l1 / l2。此方法可用于检查电池或判断电池电动势是否下降。
7. Calibration with a Standard Cell | 用标准电池校准
Calibration involves adjusting the driver circuit so that the voltage across the whole wire is a known value. A standard cell εs is used because its e.m.f. is highly stable. The standard cell is connected with its positive terminal to the high-potential end A of the wire. The jockey is tapped lightly to find the null point; the cell must never deliver significant current. Once ls is recorded, the potential gradient is firmly determined. After calibration, the standard cell is disconnected, and the unknown cell replaces it without disrupting the driver current.
校准过程调节驱动回路，使整条电阻丝两端的电压成为一个已知值。使用标准电池εs是因为它的电动势极为稳定。标准电池的正极连接到电阻丝电位较高的一端A。游标轻轻点击导线以找到零偏转点；标准电池绝不能输出大电流。一旦记录下ls，电势梯度就被确定为k = εs / ls。校准后，断开标准电池，接入待测电池，过程中不能改变驱动电流。
8. Determination of Internal Resistance of a Cell | 测量电池内阻
The potentiometer can also be used to find the internal resistance r of a cell. First, the cell's e.m.f. ε is measured on open circuit, giving balance length l1 (so ε = k l1). Next, a known resistance R is connected across the cell's terminals, and the new balance length l2 is found. Now the potentiometer measures the terminal voltage V = k l2 Since V = ε - I r and the current I = V / R, we have ε = V + (V / R) r. Rearranging gives:
r = R × (l1 / l2 - 1)
The internal resistance is thus obtained directly from the two balance lengths and the known load resistance. It is crucial that the driver current remains unchanged throughout.
电位计还可用来测定电池的内阻r。首先在开路状态下测量电池的电动势ε，得到平衡长度l1（即ε = k l1）。然后在电池两端并联一个已知电阻R，再次测量得到新的平衡长度l2，此时电位计测得的是端电压V = k l2。因为V = ε - I r，且电流I = V / R，代入可得ε = V + (V / R) r。整理后得到：
r = R × (l1 / l2 - 1)
因此，内阻可直接由两个平衡长度和已知的负载电阻求出。整个过程必须确保驱动电流保持恒定。
9. Sensitivity of the Potentiometer | 电位计的灵敏度
Sensitivity is the change in balance length per unit change in the measured e.m.f. It is inversely proportional to the potential gradient: a low k gives a large shift in the null point for a small voltage change. To increase sensitivity, a long wire can be used, or the driver voltage can be reduced (keeping it larger than all test e.m.f.s). Using a sensitive galvanometer also helps detect minute deviations from balance. However, if the potential gradient is too low, the balance point becomes hard to locate swiftly.
灵敏度指每单位被测电动势变化所引起的平衡长度变化。它与电势梯度成反比：低k值意味着微小的电压变化就能引起较大的平衡点移动。提高灵敏度可采用更长的电阻丝，或降低驱动电压（但仍须大于所有被测电动势）。使用灵敏电流计也有助于检测微小的失衡。不过，如果电势梯度过低，快速找到平衡点会变得困难。
10. Common Errors and Precautions | 常见误差与预防措施
Non-uniformity of the wire: Variations in cross-section or resistivity cause the potential gradient not to be constant. Using high-quality, carefully manufactured wire minimises this.
Contact resistance: Dirt or oxide at the jockey contact can introduce unwanted voltage drops. Clean the wire and jockey tip before use.
End errors: The scale zero may not correspond exactly to the start of the uniform part of the wire; a correction may be needed.
Heating effect: The driver current heats the wire, altering its resistance. Avoid excessively large currents and allow the circuit to stabilise.
Standard cell misuse: The standard cell must never be loaded; otherwise its e.m.f. may drift. Use a protective high resistance in series with the galvanometer during initial balancing.
电阻丝不均匀：截面积或电阻率的变化会导致电势梯度不恒定。使用高质量、精心制造的电阻丝可最大限度地减小这种影响。
接触电阻：游标接触点的污垢或氧化物会引入额外的电压降。使用前应清洁电阻丝和游标尖端。
末端误差：标尺零点与电阻丝均匀部分的实际起点可能不重合，可能需要进行修正。
热效应：驱动电流使电阻丝发热，导致电阻改变。应避免使用过大的电流，并让电路达到热稳定。
标准电池误用：标准电池绝不能输出电流，否则其电动势可能漂移。初调平衡时应在电流计回路中串接一个保护高电阻。
11. Advantages and Limitations | 优势与局限性
The potentiometer offers extremely high accuracy because it relies on a null measurement, eliminating errors due to meter internal resistance or loading effects. It can measure very small e.m.f.s and does not draw current from the source. However, it is not suitable for measuring alternating voltages directly, requires a stable driver source, and is less convenient than digital voltmeters for routine measurements. Additionally, obtaining a precise balance point demands patience and skill.
电位计具有极高的精度，因为它基于零测量原理，消除了电表内阻或负载效应带来的误差。它可以测量非常小的电动势，且不从电源吸取电流。但是，它不适于直接测量交流电压，需要一个稳定的驱动电源，并且在常规测量中不如数字电压表方便。此外，获得精确的平衡点需要耐心和技巧。
12. Summary of Key Points | 要点小结
The potentiometer balances an unknown p.d. against the known voltage drop along a uniform wire. The balance condition relies on a galvanometer showing zero deflection, ensuring no current is taken from the source. Calibration with a standard cell yields the potential gradient, enabling absolute e.m.f. measurement.
εx = εs (lx / ls)
Internal resistance r is found from the drop in terminal voltage when a load is applied:
r = R (l1 / l2 - 1)
Mastering potentiometer circuits equips students with a deep understanding of null methods, potential dividers, and careful experimental technique.
电位计将未知电势差与均匀电阻丝上的已知电压降进行平衡。平衡条件依靠电流计指零，确保不从被测电源吸取电流。使用标准电池校准得出电势梯度后，便可进行绝对电动势测量。
εx = εs (lx / ls)
内阻r则通过加载已知电阻后测量端电压的降低来求得：
r = R (l1 / l2 - 1)
掌握电位计电路能让学生深刻理解零位法、分压器原理以及严谨的实验技术。
Published by TutorHao | Physics Revision Series | aleveler.com
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