21.6 DEFINITION OF AMPERE: CURRENT BALANCE
21.6.1 The Ampere
1 Defining the unit of ampere
Consider two infinitely long, straight, thin conductors in free space which are held parallel to each other and separated by distance d. Current I1 flows in one conductor and current I2 flows in the other. Then each conductor is acted on by a magnetic force. The magnitude of the force acting on one metre of each conductor is given by
$\frac{F}{L} = \frac{\mu_0I_1I_2}{2\pi d}$
Let $I_1=I_2=I$ and $d = 1.0 \ m$. Then the force acting on unit length of each conductor is given by
$\frac{F}{L} = \left(\frac{\mu_0}{2\pi }\right)I^2$
Suppose we allow $\frac{\mu_0}{2\pi}$ to take the value
$\frac{\mu_0}{2\pi}=2 \times 10^{-7}$
and also we allow $\frac{F}{L}$ to be assigned the value
$\frac{F}{L} = 2 \times 10^{-7}$
Then we get
$2 \times 10^{-7} = (2 \times 10^{-7})I^2$
or,
$I^2 = 1$
$I = 1 \ ampere$
Hence, we can use the force between two parallel current-carrying conductors to define a current of one ampere. It is defined in the following way:
One ampere is defined to be the constant current which, flowing in each of two infinitely long, straight, parallel conductors of negligible cross-section, separated by a distance of 1 metre in vacuum, would produce a force on each conductor of magnitude $2 \times 10^{-7} \ N \ m^{-1}$.
2 A value for μ0
We have allowed $\frac{\mu_0}{2\pi}$ to take the value
$\frac{\mu_0}{2\pi}=2 \times 10^{-7}$
Hence we have fixed a value for $\mu_0$, i.e.,
$\mu_0 = 4\pi \times 10^{-7}$
EXAMPLE 21.15
Determine the units of $\mu_0$ in terms of base units.
Answer
$\frac{F}{L} = \left(\frac{\mu_0}{2\pi }\right)I^2$
$\mu_0 = 2\pi \left(\frac{F}{LI^2}\right)$
Units of $\mu_0$ are N m−1 A−2 ⇒ (kg m s−2) m−1 A−2 ⇒ kg s−2 A−2
21.6.2 The Current Balance
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| Figure 21.35 A current balance setup consisting of a U-shaped conductor placed inside a solenoid, a rod with a scale pan, and a pivot system used to measure electric current. |
One form of a current balance is shown in Figure 21.35. It consists of
(a) a light U-shaped stiff bare conductor MPQN placed inside a solenoid
(b) a light non-conducting rod RS on which we can place a light weight W
(c) a solenoid inside which can be found a uniform magnetic field when current flows in the solenoid
The light framework consisting of the rod RS and the U-shaped conductor MPQN is pivoted without friction at M and N. A voltage supply is connected to the U-shaped conductor via M and N.
The current balance can be used to carry out an absolute measurement of electric current. The measurement of current that involves only the measurement of base quantities like mass, length and time is known as absolute measurement.
21.6.3 Absolute Measurement of Current
Method:
1 Measure the length L of the portion PQ of the U-shaped conductor MPQN.
2 Insert PQ into the middle portion of the solenoid in such a way that PQ is perpendicular to the axis of the solenoid.
3 Connect PQ and the solenoid in series to form a circuit so that the same current can flow through both of them. Then connect this circuit to a voltage supply.
4 Hang a small light balance scale pan on the rod RS. Without any current flowing in the circuit, adjust the position of the pan on the rod until the framework consisting of the rod RS and MPQN is horizontal.
5 Switch on the voltage supply so that a current I flows through both PQ and the solenoid. The direction of current flow must be such that the magnetic force must pull PQ downwards.
6 Place a small mass of weight W on the scale pan and adjust the position of the pan until the rod RS becomes horizontal again.
Principle
Suppose current I flows in the solenoid and PQ. The framework consisting of the rod and MPQN is maintained by vertical forces in a horizontal position in static equilibrium, as shown in Figure 21.35.
Let m = mass of the weight on the scale pan
x = distance between M and P (N and Q)
d = distance between scale pan and pivots MN
L = length of conductor PQ
Take moment at M. We get
$Fx = mgd$
where F is the magnetic force acting on PQ. The force is given by
$F = BIL$
where B is the strength of the magnetic field inside the solenoid produced by current I flowing through the solenoid. But inside the solenoid we have
$B = \mu_0 nI$
where n is the number of turns per unit length of the solenoid. Hence we have
$(\mu_0 nI)(IL)x = mgd$
$\boxed{I = \sqrt{\frac{mgd}{\mu_0 nLx}}}$
Notice that the current I can be measured in terms of base quantities like mass (m), length (L, d, x, n) and time (second in g).
Applications of Definition of Ampere Using Current Balance and Absolute Measurement
The definition of ampere using current balance and absolute measurement has important applications in physics and electrical engineering. It provides a fundamental way to measure electric current based on force and basic physical quantities.
1. Standardization of Electric Current
The ampere defined through current balance ensures a precise and universal standard for measuring electric current in scientific and industrial applications.
2. Calibration of Measuring Instruments
Devices such as ammeters and galvanometers are calibrated using the principles of current balance to ensure accurate current measurements.
3. Fundamental Physics Experiments
The concept is used in laboratory experiments to verify relationships between magnetic force, current, and distance between conductors.
4. Development of Electrical Standards
It plays a key role in establishing international electrical standards and maintaining consistency in measurements across different systems.
5. Research and Metrology
Absolute measurement methods are important in metrology for defining and refining fundamental physical constants and units.
These applications highlight the importance of the ampere definition in ensuring accuracy, consistency, and reliability in electrical measurements.
Conclusion
The definition of ampere using current balance provides a clear and fundamental way to measure electric current based on magnetic force between conductors. By relating current to basic physical quantities such as mass, length, and time, it allows for absolute measurement without relying on comparison methods.
This concept is essential in physics and electrical engineering, as it ensures accuracy and consistency in current measurement. It also forms the basis for calibration, standardization, and the development of reliable electrical systems.
Frequently Asked Questions (FAQ) – Definition of Ampere Using Current Balance
What is the definition of one ampere?
One ampere is defined as the constant current which, flowing in two parallel conductors 1 meter apart in vacuum, produces a force of 2 × 10-7 N per meter between them.
What is current balance?
Current balance is an instrument used to measure electric current by balancing magnetic force with gravitational force.
What is absolute measurement of current?
Absolute measurement of current is a method of measuring current using only base physical quantities such as mass, length, and time.
Why is the definition of ampere important?
It provides a standard and accurate way to measure electric current, ensuring consistency in scientific and engineering applications.
What is the role of μ₀ in this definition?
μ₀ (permeability of free space) determines the relationship between current and magnetic force in vacuum.
How does current balance work?
It works by balancing the magnetic force produced by a current-carrying conductor with a known weight, allowing the current to be calculated.
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