📚 Potentiometer Circuits | 电位差计电路
In CIE A Level Physics, a potentiometer is a slide-wire instrument used to compare or measure electromotive force (emf) and potential difference by a null method. It is one of the most accurate ways to measure the emf of a cell because it draws no current from the cell at balance.
在 CIE A-level 物理中，电位差计是一种滑线式仪器，通过零平衡法比较或测量电动势和电势差。它是测量电池电动势最准确的方法之一，因为平衡时不从电池吸取电流。
1. What is a Potentiometer? | 什么是电位差计？
A potentiometer in this topic is not the small three-terminal variable resistor used in volume controls. It is a measuring circuit consisting of a long uniform resistance wire, a driver cell, a sliding contact called a jockey, and a sensitive galvanometer. The wire is usually one metre long and made of an alloy such as constantan or nichrome.
本主题中的电位差计不是音量旋钮中那种三端可变电阻器。它是由一根长而均匀的电阻丝、一个驱动电池、一个称为滑动触点的滑动接头和一台灵敏检流计组成的测量电路。电阻丝通常长一米，由康铜或镍铬合金等合金制成。
The key idea is that when a steady current flows through a uniform wire, the potential drop along the wire is proportional to the distance along the wire. A balance point can therefore be used to compare an unknown emf with the potential drop across a measured length of wire.
核心思想是：当稳定电流通过均匀电阻丝时，沿线电势降与沿线距离成正比。因此，可以利用平衡点将未知电动势与电阻丝上某一段长度两端的电势降进行比较。
2. Basic Circuit and Null Method | 基本电路与零平衡法
The driver cell sends a steady current through the wire AB, producing a potential drop from A to B. The unknown cell is connected with the galvanometer to the jockey J. The jockey is slid along the wire until the galvanometer reading is exactly zero. This is called the balance point or null point.
驱动电池使稳定电流通过电阻丝 AB，从 A 到 B 产生电势降。待测电池与检流计一起连接到滑动触点 J。滑动触点沿电阻丝移动，直到检流计读数恰好为零。这一点称为平衡点或零点。
At balance, the potential difference across the length AJ is equal to the emf of the unknown cell. Since no current flows through the galvanometer branch, no current is drawn from the unknown cell, and the measured value is its true emf rather than its terminal potential difference.
平衡时，长度 AJ 两端的电势差等于待测电池的电动势。由于没有电流流过检流计支路，待测电池不输出电流，因此测得的是它的真正电动势，而不是端电压。
E = V_AJ = k l
Here E is the unknown emf, V_AJ is the potential difference across length AJ, k is the potential gradient, and l is the balance length.
式中 E 为未知电动势，V_AJ 为长度 AJ 两端的电势差，k 为电势梯度，l 为平衡长度。
3. Potential Gradient Along the Wire | 沿电阻丝的电势梯度
If the wire is uniform and the driver current I is constant, the potential falls at a constant rate along the wire. The potential gradient k is the potential difference per unit length of the wire.
如果电阻丝均匀且驱动电流 I 恒定，则沿线电势以恒定速率下降。电势梯度 k 就是电阻丝上单位长度的电势差。
k = V_AB / L = I R_AB / L = I ρ / A
In this expression, V_AB is the total potential difference across the full length L of the wire, R_AB is the resistance of the wire, ρ is its resistivity, and A is its cross-sectional area. The SI unit of potential gradient is V m⁻¹.
式中 V_AB 是电阻丝全长 L 两端的总电势差，R_AB 是电阻丝的电阻，ρ 是其电阻率，A 是其横截面积。电势梯度的 SI 单位是 V m⁻¹。
For accurate work, the driver current must remain constant during the experiment. If the current changes, the potential gradient changes, and all balance lengths change in the same proportion.
为了准确测量，驱动电流必须在实验过程中保持不变。如果电流变化，电势梯度就会变化，所有平衡长度都会按相同比例变化。
4. Comparing Two EMFs | 比较两个电动势
To compare two emfs E₁ and E₂, connect the first cell and record its balance length l₁. Then connect the second cell without changing the driver circuit and record its balance length l₂.
为了比较两个电动势 E₁ 和 E₂，先接入第一个电池并记录其平衡长度 l₁。然后在不改变驱动电路的情况下接入第二个电池，记录其平衡长度 l₂。
Because the same potential gradient k is used for both readings, the emfs are proportional to their balance lengths.
因为两次读数使用相同的电势梯度 k，所以电动势与它们的平衡长度成正比。
E₁ / E₂ = l₁ / l₂
This result is very important in CIE questions. The ratio of emfs is equal to the ratio of the corresponding balance lengths, provided the driver current and wire temperature do not change.
这个结果在 CIE 考题中非常重要。只要驱动电流和电阻丝温度不变，两个电动势之比就等于对应的平衡长度之比。
5. Measuring an Unknown EMF | 测量未知电动势
To measure an unknown emf accurately, first use a standard cell with a known emf Eₛ. Find its balance length lₛ and calculate the potential gradient.
为了准确测量未知电动势，首先使用已知电动势为 Eₛ 的标准电池。找到它的平衡长度 lₛ，并计算电势梯度。
k = Eₛ / lₛ
Then replace the standard cell with the unknown cell and find its balance length lₓ. The unknown emf is given by:
然后用待测电池替换标准电池，找到其平衡长度 lₓ。未知电动势由下式给出：
Eₓ = k lₓ = Eₛ × (lₓ / lₛ)
This method is direct and does not require knowing the driver current or the resistance of the wire. Only the ratio of lengths and the known standard emf are needed.
这种方法直接明了，不需要知道驱动电流或电阻丝的电阻。只需要长度之比和已知的标准电动势。
6. Measuring 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, connect the cell alone and find the balance length l₁ when no external resistor is connected. Since no current is drawn, this gives the emf:
电位差计也可用于测量电池的内阻 r。首先单独接入电池，在没有外接电阻时找到平衡长度 l₁。由于不吸取电流，因此得到电动势：
E = k l₁
Then connect a known resistor R across the cell. The cell now supplies a current, and its terminal potential difference V is measured at the new balance length l₂:
然后在电池两端并联一个已知电阻 R。此时电池输出电流，在新的平衡长度 l₂ 处测得的是端电压 V：
V = k l₂
For a cell of emf E and internal resistance r supplying a load R, the terminal potential difference is V = E R / (R + r). Substituting the two balance equations gives:
对于电动势为 E、内阻为 r 的电池向负载 R 供电，其端电压为 V = E R / (R + r)。代入两个平衡方程可得：
r = R × (l₁ – l₂) / l₂
| Symbol | Meaning | 中文含义 |
| l₁ | balance length with no external resistor | 无外接电阻时的平衡长度 |
| l₂ | balance length with resistor R connected | 接入电阻 R 后的平衡长度 |
| R | known load resistance | 已知负载电阻 |
| r | internal resistance of the cell | 电池内阻 |
This calculation assumes that the potential gradient k remains constant throughout both readings.
该计算假设两次读数过程中电势梯度 k 始终保持不变。
7. Calibrating the Wire and Checking the Driver Current | 校准电阻丝与检查驱动电流
Because the potential gradient depends on the driver current, it is good practice to calibrate the wire using a standard cell before and after the main measurements. If the standard cell balance length changes, the driver current has drifted, and the readings may be unreliable.
由于电势梯度取决于驱动电流，最好在主要测量前后使用标准电池校准电阻丝。如果标准电池的平衡长度发生变化，说明驱动电流发生了漂移，读数可能不可靠。
Common causes of drift include the driver cell discharging, the wire heating up, or poor connections changing the total circuit resistance. Waiting for thermal stability and tightening all connections improves reproducibility.
引起漂移的常见原因包括驱动电池放电、电阻丝发热或接触不良改变电路总电阻。等待热稳定并拧紧所有连接可以提高重复性。
8. Sensitivity, Precision and Precautions | 灵敏度、精度与注意事项
The precision of a potentiometer measurement depends on how sharply the galvanometer null can be located. A long wire, a sensitive galvanometer, and a clean uniform wire all improve precision.
电位差计测量的精度取决于检流计零点的定位是否敏锐。长电阻丝、灵敏检流计和干净均匀的电阻丝都能提高精度。
Near the balance point, the galvanometer deflection is very small. Before finding the balance point, a protective resistor should be connected in series with the galvanometer to limit current and avoid damage. The protective resistor can be shorted out or reduced as the balance point is approached.
在平衡点附近，检流计偏转非常小。在寻找平衡点之前，应在检流计上串联一个保护电阻，以限制电流并避免损坏。接近平衡点时，可将保护电阻短接或减小。
- Use a steady driver cell and avoid a large current that heats the wire.
- 使用稳定的驱动电池，避免大电流使电阻丝发热。
- Check all contacts and clean the wire if necessary.
- 检查所有触点，必要时清洁电阻丝。
- Record balance lengths in the
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