kinematics  
 
  Eddy currents  
 
  Lenz's law  
 
  Lines of fields  
 
  Insulators  
 
  Diff. Eqs   
 
  Constants  
 
  Units   
 
  home  
 
  ask us  
 

 








© The scientific sentence. 2010

DC current Electric energy and power




1. Power dissipated in a dc circuit element

The electric energy in a circuit supplied by an electric source is used by the elements in the circuit. The element in the circuit transforms the energy received by decreasing the potential through this element.

If a current passes through the element ab, the potential decreases, the potential difference Vb - Va is negative.

In a time interval Δt, the number of charge carriers with total charge ΔQ that enters the element at the terminal a of potential Va is the same as that leaves the element at the terminal b of potential Vb. That is the current I = ΔQ/Δt is constant.

The electric potential energy at the terminal a is Ua = ΔQ Va, and the terminal b is Ub = ΔQ Vb.

The change in the electric potential energy is ΔU = Ub - Ua = ΔQ (Vb - Va) = - ΔQ (Va - Vb). Va - Vb is positive and denoted by V. Hence
ΔU = - ΔQ V

The rate at which the carriers lose electric potential energy is - ΔU/Δt. This rate is called the power dissipated in the element. It is equal to = + ΔQ/Δt V = VI, therefore.


Power dissipated in a dc circuit element:

P = V I

The dimension of V is the Volt (V), that is the energy per unit charge; and the dimension of I is the Ampere (A), that charge per unit time. Hence their product has the dimension of V A, that is (energy/charge) x (charge/time) = energy/time, that is power.



2. Energy dissipated in a resistor



The energy transformed by a resistor of resistance R, when it carries a current I. The charge carriers lose electric potential energy when they pass through this resistor. The related dissipated power is

PR = V I = R I I = R I2, called Joule's law.

Since V = R I, we have also PR = V2/R.


Joule's law for a resistor of resistance R in a dc circuit element:

PR = R I2 = V2/R

The lost electric potential energy of the charge carriers in the resistor is due chiefly to the collisions of the carriers within the resistor while the resistor is carrying a current. This is what makes resistance for a resistor. These collisions are responsible for the increase of the temperature in the resistor. The resistor transfers heat to its surroundings if its temperature rises above that of its surroundings. The carriers lose energy that is transferred as heat to the surroundings. Electric energy is dissipated as heat in a resistor, called Joule heating.



3. Energy in a battery

We want to determine the related transformation of energy in a battery while a current exists in the battery.


3.1. Discharging battery: Energy from a battery

Consider a discharging battery. The charge carriers pass through the battery in the direction of its emf ℰ. The potential across the battery increases, hence the potential difference across the battery is positive. The rate at which the carriers gain potential energy is + ΔU/Δt.

The potential difference V between the terminals of the battery is V = ℰ - r I (r is the internal resistance of the battery, and I is the current that passes through the discharging battery. The battery is being discharged,hence the energy is supplied; we call the gained potential energy + ΔU/Δt, the power output Po from the battery. Therefore

Po = VI = (ℰ - r I) I = ℰI - rI2


Power output from a discharging battery:

Po = VI = (ℰ - r I) I = ℰI - rI2 = P - Pr


The term P = ℰI represents the rate at which the electric potential energy of the carriers is increased by chemical reactions in the battery, that is the power expended by the emf of the battery.

The term Pr = rI2 is the Joule heating or the energy dissipated as heat due to the internal resistance r of the battery. While it is discharging, the temperature of the battery increases.



3.2. Charging battery: Energy to a battery

If a battery is being charged, then the sense of the current is opposite the sense of the battery's emf. Hence V = ℰ + r I, and Pi = VI = (ℰ +r I) I = ℰI + rI2 = P + Pr. Pi is called the Power input to a charging battery. P represents the power delivered to the emf of the battery by the charge carriers.



4. Energy in a circuit element

In any type of circuit element, if V is the potential difference across the element and I is the current in the element, Then the rate of energy transformation is P = V I. It is the rate at which the electric potential energy of the carriers changes as they pass through the element.


Power of any circuit element = Rate at which energy is transformed in a circuit element:

P = VI







 


chimie labs
|
Physics and Measurements
|
Probability & Statistics
|
Combinatorics - Probability
|
Chimie
|
Optics
|
contact
|


© Scientificsentence 2010. All rights reserved.