Showing posts with label Current Electricity. Show all posts
Showing posts with label Current Electricity. Show all posts

Force between two infinitely straight long conductors

Let us consider two infinitely straight long conductors carrying current in the same direction.We can measure the force between the two conductors as shown below. We need to use the formula that we have derived the formula for the magnetic field at a given point using the Ampere’s law.



Basing on the magnetic field, we have derived for the force experienced by the point at a distance using the formula that was also derived.

The second conductor also carrying current and experience due to the other conductor similar to the first conductor. These forces are mutual and we can derive as shown below.


Definition of ampere

We can  define  the ampere basing on the derivation as derived earlier. If two infinitely long straight conductors carrying a certain  current separated by  unit distance experience a force of repulsion per  unit length, the current passing through each conductor is one ampere.

This is treated like a fundamental unit in the SI system  and is defined  As shown below.




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Force on Current Carrying Conductor Fleming Left Hand Rule

We can find the magnetic field around a wire on the axial line of charge using the Biot-Servert’s law. We are not adding the derivation here and the expression is as shown in the figure.

Let us consider a point on the axial line at a particular distance and let us assume that we know the radius at any given point.



The expression for the magnetic field at any point can be expressed as shown in the diagram. If we are measuring it at the center of the circle, we need to equate the value to zero.


We can also find out the force acting basing on Biot-Servert’s law. We can find the magnetic field at any given point using this rule. We know that the magnetic induction is defined as the force experienced by a unit north pole when placed in a magnetic field.


Thus we can measure the force as the product of the pole strength and magnetic induction. As the field is small component, the force is also small component. To get the total force acting on the point, we need to integrate the given equation and we can get the total force as shown in the diagram below.


This force will be maximum when the point is perpendicular to the current carrying conductor. If  the angle is zero or 180 degree, the force will become zero as shown below.


To  find  the  direction of the force experienced  by  the current carrying conductor using Fleming Left hand rule. As per the law, if  fore finger  indicate the direction of the magnetic  field and central finger indicates  the direction  of the  current then the thumb  indicates the direction of the thrust or force experienced by the current carrying conductor.

We  can also measure the force acting on  a charge simply by defining  the current as the rate of charge. We can define as the cross product of velocity of the charge and the magnetic field and the product is multiplied  with the charge.


We can  define  the  unit of  magnetic  induction tesla basing  on the above derivation. The magnetic  field induction is the force experienced by the  conductor when a  unit charge passing through  a conductor with unit velocity  at right angle.



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Amepere's Law and Magnetic Field around Conductor

We can also find out the magnetic field induction at any point due to a charge using the Ampere’s law.According to this rule the line integral of magnetic induction around a closed curve is permittivity of free space times the current in that closed loop.


Problem and solution

Let us consider a current carrying conductor in circular shape and we are interested in the magnetic field at the center of the coil. We can use the formula that we have derived to do that and we shall assume that the distance of the particle on the perpendicular axis is zero. It is because we are measuring it at the center of the coil. The problem is solved as shown below.



When we measure the line integral, we get the length of the wire around which we are measuring the magnetic field. We also need to measure the magnetic field only due to currents inside the closed loop. We need not worry about the currents outside as they do not produce any impact. We are measuring only due to the portion of currents that are in the closed loop.

The currents with in the loop which are coming into the loop are treated as positive and currents leaving the closed circuit shall be treated as negative.



Basing on this Ampere’s law, we can find the magnetic field around a closed straight current carrying conductor of infinite length as shown below.

Let us assume a conductor carrying a current “I” as shown in the figure. We would like to measure the magnetic field around it at a distance “r” from it.  We can consider the line integral around it as the circular path of the given radius and when we line integrate it; we get the length of that closed path. It is nothing but the circumference of the circle.



It is the dot product of the magnetic field and the component of the length due to which we need to measure the field as per the Amper’s law. Any way the field and the portion of the length are in the same direction and the angle is treated as zero.

In the place of that line integral of the component of the length, we need to write the circumference as shown and we can find the magnetic field as shown below.



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Magnetic Field due to Current Carrying Conductor

In the beginning electricity and magnetism were treated like different subjects. People could not find any relation between them. Electricity is found to be due to the charges and magnetism is thought to be due to the different poles like north and south poles. The force between them and treatment of each subject is done quite separately and they do exist like to separate branches of physics.

In the earlier days of nineteenth century some experiments done by famous scientists found that with the change in the electric field, there is magnetic field also developed around it. This leads to new science called electromagnetism.

The experiments found that the charged particle in the state of rest gives electric field around it. If the charge is moving or if there is a flow of current in any conductor, around it there is not only electric field and there is also a magnetic field.

It can be noticed that the magnetic needle with north and south poles will keep on changing its direction around a electric charge as shown in the diagram.



We can find the direction of the magnetic field using different laws. One of those kinds of rule is Maxwell’s cork screw rule. If there is a nail that rotates using the right hand and if we rotate the screw in such a way that the nail advances in the direction of the current, the direction of the magnetic field is along the direction of the rotation of the head of the nail. It can be understood that the rotation of the head of the nail and the tip of the nail are in the perpendicular plane. Thus electric and magnetic fields are in the perpendicular plane.


We can also use a rule called right hand thumb rule to identify the direction of the magnetic field. It is somehow similar to cork screw rule. If we hold a current carrying conductor with our hand such that the thumb is along the direction of the current, the direction of the magnetic field is along the direction of the curled fingers. It again tells you that the electric field and the magnetic field are in the perpendicular planes.


Around every magnetic pole, there is some space up to where its influence can be experienced. That space is called magnetic field. If we keep any other magnetic pole with in that field, it experiences a force of attraction or repulsion. That force is called magnetic field induction. We can define the magnetic field induction as the force experienced by a unit North Pole placed in the magnetic field.

Around every current carrying conductor, there is a magnetic field and there is a magnetic field induction. To measure that value we have different rules and one among them is Biot-Servert’s law. According to this rule, the magnetic induction at any point directly proportional to the some factors like the  SIN angle it makes with the point, current passing in the conductor, component of the part of the length of the portion of the wire due to which we are measuring the magnetic field and is inversely proportional to the square of the distance of separation.




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Current Electricity Complete Lesson

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Current Electricity Complete Lesson

Current electricity is a branch of physics that deals with the charges in motion and its applications. Current flows through the conductors and while it is happening some opposition is there called resistance. In this lesson we have analyzed on whom this resistance is depending on and how can we measure it. To measure the nature of resistance and its dependence, a physical quantity called specific resistance is also defined. It depends on the nature but not on the physical dimensions of a body. 

To know the dependence of the current on potential difference in simple cases, Ohm's law is defined and to study the complex cases, we have Kirchhoff's laws. To know about the impact of resistance, we have Wheatstone bridge and its application Meter bridge. We also deal here about potentiometer and it is useful to compare the EMF of different cells and to find the internal resistance of a battery. Detailed lessons were made about each of the above topics and they are listed here for the reference.

Resistance and Specific Resistance

EMF and Internal Resistance of a Cell

Kirchhoff's Laws and Explanation

Kirchhoff's Law Problems and Solutions

Wheatstone bridge and Meter Bridge

Potentioemeter Comparison of EMF's and Determination of Internal Resistance

Resistors in Series and Parallel Problems and Solutions


Other complete lessons in this blog are mentioned here for the reference.

Gravitation Complete Lesson

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Resistors in Series and Parallel Problems and Solutions

Resistors in Series

Resistors in series means end to end connection. When they are connected in series, we can find that the current will be the same across all of them and the supplied voltage across them is distributed proportional to the resistance.

We can find the effective resistance of the system when number of resistors are connected in series and also find out the voltage drop across each resistors as shown below. The effective resistance increases in series combination.



This concept is explained here in this video for your reference.

resistors in series

Resistors in parallel

When similar end of all resistors are connected together and the same with the other ends, the connection is called parallel connection. In parallel connection, the voltage across all the elements is same and the current across them is distributed such that it is inversely proportional to resistance.

We can find the effective resistance and the current in each element as shown below. The effective resistance in parallel is less than even the small value of the circuit. The effective resistance decreases in parallel combination.



In a video lesson over you tube parallel resistence resultant is found as shown in the link below.

Parallel combination of resistors

Variation of resistance Problems and Solutions

We need to find the effective resistance between the two points as shown in the given picture. We can simply solve the problem by bisecting the entire circuit into two identical parts. The two parts are in series with each other and symmetrical. If we are able to find the resistance of one part, by adding the same value to that, we can find the total resistance of the system given.

As we have bisected the resistor into two parts, its length so its resistance also becomes half. By identifying the resistors in series and parallel and measuring their effective resistance, we can find the total resistance of the circuit as shown below.


Problem and Solution

This problem is about percentage change in the resistance of a wire when there is a change in its length alone. No information is given in the problem about its area. As the volume of the wire remains constant, we need to write area in terms of length and volume. Thus we can prove that resistance is directly proportional to the square of the length of the wire.

The problem is solved as shown below.


Problem and solution

This problem is about variation of resistance with mass of the wire. There is information in the problem about its length but not about area. We can change the area interms of mass and solve the problem as shown in the diagram below.


Problem and solution

This problem is about current passing in a wire when multiple wires are connected in parallel. The total current across the combination is given to us and resistance of each wire is given. We know that if resistors are connected in parallel, the voltage across them is same. Hence the current flow is reciprocal to resistance and the problem can be solved as shown below.


Problem and solution

This problem is about finding a voltage across a resistor when multiple resistors are connected in the circuit as shown in the diagram.

By identifying the elements and currents across them, we can solve the problem as shown below.


Problem and solution

If two parts of a circle separated by an angle are two wires having different resistance, we need to measure the effective resistance of the circuit. The problem is solved as shown below.


Problem and solution

This problem is about to find the effective resistance of the system where infinite resistors are connected as shown in the diagram. We can identify the symmetry in the ladder and we can say that the circuit is the combination of similar symmetrical parts. Except one part, we can assume that all other parts together are having some resistance and even with the other remaining ladder, the answer still remains same. Simply because of infinite ladders, adding or removing one ladder is not going to make a big difference to the entire system significantly.

Thus we can solve the problem as shown below.




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Capacitors in Series and parallel with Problems and Solutions
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Potentiometer Comparison of EMF's and Determination of Internal Resistance

Potentiometer is a device used to compare the Emf’s of two cells and also used to find the internal resistance of a given cell.It works on a principle that the potential drop across a wire is directly proportional to the length of the wire through which the current is passing.




The device consists of two parts called primary and secondary circuit. The primary circuit consists of a strong battery to supply the potential requirement of the circuit. The connection from the cell is given to a long wire. This long wire is divided into pieces of wires each of one meter length and they are connected with small copper pieces of negligible length. They behaves like multiple wires connected in series and the total length is available between the copper strips.A galvanometer is connected from the wire with a connecting wire.

Comparison of EMF's of two cells

The secondary circuit consists of cells whose electromotive forces has to be compared. There is key that acts like a on and off switch and it can be used to control the flow of current through a particular cell.


In two different cases the two cells are connected in the secondary circuit separately and corresponding balancing lengths are measured with it. When the emf of secondary circuit is balanced with that of primary, the galvanometer shows zero deflection. This happens at a particular length of the wire and that length is called balancing length. We will find the balancing length with the second cell whose EMF has to be compared.

As per the principle, the ratio of EMF’s is nothing but the ratio of the balancing lengths in the two cases. Thus we are able to compare the EMF’s of two cells using the Potentiometer.

Determination of internal resistance of a cell

We need to connect cell alone first whose internal resistance has to be measured. We can find the balancing length in the circuit. In the next case, the same cell is connected to a external resistance in parallel and again balancing length is measured using the potentiometer. There we will get one more balancing length. The ratio of emf of that cell and its potential drop across a resistor is nothing but the ratio of balancing lengths.

Basing on the definition of potential drop, we can get one more relation between emf and voltage and by comparing both of them, we can measure the internal resistance of a cell as shown below.




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Working of Capacitor and Dielectric effect on Capacity
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Wheatstone bridge and Meter Bridge

Wheatstone bridge

Kirchhoff laws are useful to find the current passing across a element in a electric circuit. We can apply first law that deals with current coming towards a junction is the current leaving the junction. The second law is about conservation of potential difference across a closed circuit. It is easy to apply this laws to a complex circuit and solve the problem.

Wheatstone bridge is a particular arrangement of four resistors with which we can measure the forth resistance if three o them are known to us.

Between two junctions we can connect a galvanometer and we can check the current passing in the circuit. Between the other two junctions, a cell with a known EMF is connected.

We can apply kirchhoff’s second law in each closed loop and find out the equations and we can solve them further. We can apply the law only when the wheatstone bridge is balanced. When it is balanced, the current passing across the galvanometer is zero. In that case the ratio of the resistors is same.



We can derive the condition for the wheatstone bridge using the Kirchhoff laws as shown below with all the conditions and sign conventions.

When the bridge is balanced, we can find as shown below that, the ratio of the resistance of the pairs is same.


Wheatstone bridge is a particular arrangement of four resistors with which we can measure the forth resistance if three o them are known to us.

Problem and Solution

The following problem is about finding the balance of Wheatstone bridge. Four resistors were given in the bridge but the bridge is not balanced. We need to know the resistance to be connected to any one resistor so that the bridge is balanced.

To solve this problem, we simply need to apply the condition for the balanced wheatstone bridge and by simplifying the equation, we can solve the problem as shown below.


Meter bridge

Meter bride is a modified version of wheatstone bridge. There is practical issues in using the wheatstone bridge. The serious problem is getting the balance of the bridge. We can apply the rule only when the circuit is balanced and the galvanometer shows zero deflection. For that to happen, we need to keep changing one resistors keeping the others same.

The balance happens at a particular resistance and we don’t know that value. Finding that value by keep changing the circuit is quire a time consuming and impractical process.

To avoid this problem, two of the resistors were replaced with a wire and jockey. It will divide the wire into two parts and each part has a particular resistance basing on its length.

We can prove that the ratio of the two resistors is the ratio of the lengths of the two parts of the meter wire taken in the experiment.

Problem and solution

We need to solve the resistance of a wire kept in the meter bridge and we are changing the temperature at the other end. So we need to take coefficient of temperature into count and find the value of resistance as shown below.




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Electric Potential Problems with Solutions
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