Category: Physics Notes PPT

An Electric Fuse is an Electric device which interrupts the flow of current in an Electric circuit. It is installed in a circuit to stop the flow of excessive current. A Fuse is usually a short piece of wire. The Fuse is made up of a material which has high resistivity and low melting point, so that it melts down due to overheating of the wire during high current flow.

The thickness of the Fuse wire is determined based on the amount of current flow in the circuit. Normally an alloy of tin and lead is used as the Fuse wire, as it has high resistivity and low melting point.

If a fault causes a flow of excess Current then a thin Conductor is used to break the Circuit by melting or separating it, the thin Conductor used is known as an Electric Fuse. A Fuse can be sacrificed if anything in the Circuit goes wrong since they are weak points that are intentionally placed in a Circuit. For example, in order to protect the wiring of the vehicles, a Fuse panel is placed near the batteries of the Car.

The wire inside the Fuse melts if there is an occurrence of high Current due to a short Circuit or an overloaded Circuit. As a result of which the Current stops flowing since the wire has broken. In order to stop the flow of Electricity, the Electric Fuse gives up its life. There is a clear plastic window in some Fuses from which one can check if they are still good.

Electric Fuse- Working Principle

The Electric Fuse works on the basis of the heating effect of the Electric Current. It is composed of a non-flammable thin metallic wire with a low melting point.

If a high amount of Electricity is passed from the Electric Fuse, there is a production of heat which causes the Fuse to melt which leads to the opening of the Circuit and the blockage of Current.

Once a Fuse melts, it can be changed or replaced with a new Fuse.

 

A Fuse is normally made up of elements like zinc, copper, aluminum and silver.

 

A Fuse acts as a circuit breaker and breaks the circuit in case any fault occurs in the circuit. It acts as a protector of Electric appliances and also as a safety measure for humans. The figure below represents a Fuse operation, Fuse barrel and Fuse link.

Characteristics of an Electric Fuse

Here are some important characteristics of a Fuse wire.

• Current Rating: It is defined as the continuous conduction of maximum current holded by the Fuse without melting. It is the capacity of current, and is measured in Amperes. Current (Cin)=75% current (rating)

• Voltage Rating: If voltage is connected in series with the Fuse, it does not increase voltage rating.

Hence,

V (Fuse) >V (open circuit)

• I2t Rating: It is the total energy which is carried by the Fuse element in case of a short circuit. It measures the heat energy of the Fuse, and is generated when the Fuse breaks out.

• Interrupting or Breaking Capacity: The maximum rating of current without harming the interruption by the Fuse is known as interrupting capacity of the Fuse.

Breaking capacity > maximum rated voltage

Breaking capacity < short circuit current

• Voltage Drop: The Fuse element melts whenever there is an excessive current in the circuit, and opens the circuit. Due to this, voltage drop and resistance change reduces.

• Temperature: The Fuse melts when the operating temperature is higher and the current rating is lower.

 

The graph represents temperature vs current carrying capacity of a Fuse. The current carrying capacity of a Fuse is 100% when the temperature is 25°C (three lines meet at this point). After that the current carrying capacity decreases upto 82% at 65°C. This shows that increase in temperature decreases the current carrying capacity of a Fuse.

Electric Fuses are proven to be helpful in safeguarding any Electrical appliance or household Circuits. Its features are-

1. The melting points of Electric Fuses are very low, which is 200° C.

2. Fuse wires are made of an alloy composed of 50% lead and 50% tin.

3. The resistance of Fuse wires is so high that whenever its temperature rises and it reaches its melting point, it breaks the Current which passes through the Circuit.

Functions of Electric Fuse

Electric Fuses have to give up their lives in order to safeguard the Circuits. Some other important functions of Electric Fuses are listed below-

1. Restricting the flow of Current- An Electric Fuse acts as a barrier between an Electric circuit and the human body.

2. Preventing the wires from catching fires or breakdowns- It prevents any damage to the Electric device by restricting excess current flow.

3. Terminating the Current from the Circuit if a short- Circuit or overloading happens- When too many appliances are connected in a single circuit, it leads to overload which requires a Fuse to terminate the circuit connection.

4. Prevention from blackouts- if any dis-function occurs in the components of the circuit, the nearest circuit breaks.

5. Prevention from damage occurred due to mismatched loads.

Information about the ampere rating, voltage rating, approval standards of the Fuse and interrupt rating are generally marked on the Fuse. This information must be checked and verified before buying a Fuse.

Amplitude refers to the maximum change of a variable from its mean value (when the variable oscillates about this mean value). In to and fro motion of a particle about a mean position, it is the maximum displacement from its mean position. Similarly, amplitudes are defined for periodic pressure variations, periodic current or voltage variations, periodic variations in electric or magnetic fields etc.

There is no particular formula for amplitude. It’s available from the equations or the graphical representations of such variations.

If y = A sin ωt ampere. 

What is the peak value of the current?

Options:

(a) 5 A

(b) 2.5 A

(c) 4.33 A

(d) 7 A

Answer: (a)

Let us understand what amplitude is before moving on to the amplitude formula. The amplitude is the maximum displacement of any particle in a medium from its state or equilibrium position. The letter ‘A’ represents amplitude. The amplitude of a bounded-range periodic function is half the distance between the minimum and greatest values. The amplitude is the distance between the centerline and the peak or trough. Let us look at the amplitude formula and solve a few examples.

What is the Amplitude Formula?

The largest deviation of a variable from its mean value is referred to as amplitude. The sine and cosine functions can be calculated using the amplitude formula. A is the symbol for amplitude. The sine (or cosine) function can be written as follows:

x = A sin (ωt + ϕ)   or   x = A cos (ωt + ϕ)

Here,

The amplitude formula is also expressed as the average of the sine or cosine function’s maximum and minimum values. The absolute value of the amplitude is always used.

Example: A wave is y = 2sin(4t). Find out its amplitude.

Solution:

Given: wave equation y = 2sin (4t)

using the amplitude formula,

x = A sin(ωt + ϕ)

When compared to the wave equation,

A = 2

ω = 4

ϕ = 0

As a result, the wave’s amplitude is 2 units.

Example: What is the intensity of light incident normal to a circular surface of radius 5 cm from a 100 W source of light?

Solution:

r = 5 cm = 5 × 10–2 m, P = 100 W, [I]=?

[I = frac{P}{{pi {r^2}}} = frac{{100}}{{pi times {{(5 times {{10}^{–2}})}^2}}} = 1.27 times {10^4}W{m^{–2}}]

Question: The intensity of a light source with power P on a circular surface of radius r placed at a certain distance is [I]. If at the same distance from another source, if the intensity on a circular surface of double the radius, is 4[I], then what is the power of the second source?

Options:

(a) 4 P

(b) 16 P

(c) 2 P

(d) P / 4

Answer: (b)

For an object exhibiting a circular motion, there are always some parameters to describe its nature. 

 

If we talk about a particle’s velocity, which is an angular velocity, that remains constant throughout the motion; however, angular acceleration makes two types of components and they are tangential and radial acceleration.

 

Tangential acceleration acts tangentially to the direction of motion of a particle and remains perpendicular to the direction of the radial component. Now, we’ll discuss the tangential and radiation acceleration formula.

 

Tangential Acceleration and Centripetal Acceleration Formula

Tangential acceleration meaning is a measure of how the tangential velocity of a point at a given radius varies with time. Tangential acceleration is just like linear acceleration; however, it’s more inclined to the tangential direction, which is related to circular motion.

 

If we talk about the narrow gap between the centripetal acceleration, which is an acceleration that acts towards the center of the circle along which the body or a particle is creating a circular motion.

 

We can see there is a narrow line of difference between the two types of acceleration, and that the difference lies in the way the acceleration acts on the particle in a circular motion.

 

Now, let’s discuss the tangential acceleration equation followed by the centripetal acceleration.

 

Tangential Acceleration Formula

Let’s suppose that you and your friends are playing with a string. You are in the middle of the string and your friends have joined the string from hand to hand and are moving with high speed or changing speed in a circular motion.

 

Here, we are talking about angular velocity and we know that change in the velocity is called acceleration, which is angular acceleration. So, we can write the first derivative of angular velocity concerning time for angular acceleration. 

 

Acceleration Tangential Formula 

Here, we aim to describe the tangential acceleration formula, so we will focus more on it, as our article relies on the same. 

 

Now, writing the tangential acceleration equation in the following manner:

    

Tangential acceleration [a_{t} = r times frac{text{d}omega}{text{d}t}]……(1)

 

So, we denote the tangential acceleration with a subscript ‘ct’ along with the English letter ‘a’. 

  Here, [frac{text{d}omega}{text{d}t}] = angular acceleration

r = is the radius of the circle

 

Since the motion talks about the position of a particular object that’s why we call ‘r’ as the radius vector.

 

We also know that the angular velocity can be written, so we can rewrite the above equation (1) to get the Tangential Acceleration Formula Circular Motion in a new form:

at = r𝛼….(2)

 

Centripetal Acceleration Formula

The centripetal acceleration of an object making a circular motion with a circle ‘r’ and having a speed ‘v’ in meter per second is given by the following centripetal acceleration equation:    

 

                      aC = v2/r

 

So, we denote the centripetal acceleration with a subscript ‘c’ along with the English letter ‘a’. 

 

Now, we will discuss the radial and tangential acceleration formula in detail.

 

Tangential and Radial Acceleration Formula 

We already discussed the acceleration tangential formula in the above context, while talking about the narrow difference between the centripetal and tangential acceleration, we also saw a minor difference between tangential acceleration and the centripetal acceleration formula.

              

Now, let’s discuss the radial acceleration:

 

Radial Acceleration

We define radial acceleration as the component that points along the radius vector, the position vector that points from a center, usually of rotation, and the position of the particle that is accelerating.

 

The formula for radial acceleration is given by: 

                   ar = v2/r …..(3)

 

Here, we can see the term ‘r’ or the radius vector has a difference in the tangential acceleration and the centripetal acceleration formula. Also, we notice that the centripetal acceleration and the radial acceleration have the same formula.

 

 

Finding Tangential Acceleration

Now, we will look at one problem to find the tangential acceleration of an object. 

 

Example: 

If an object is experiencing a circular motion, then what will be its tangential acceleration? Also, determine the overall acceleration of the object.

 

Answer: 

The overall acceleration of an object is given by the following equation:

 

[vec{a_{(total)}} = vec{a_{(r)}} + vec{a_{(t)}}]

Now, tangential acceleration can be determined by subtracting the radial component acceleration from the overall acceleration in the following manner:   

 

 [vec{a_{(t)}} = vec{a_{(total)}} – vec{a_{(r)}}] 

 [vec{a_{(t)}} theta cap = vec{a_{(total)}} – vec{a_{(r)}} r(cap)]

If we wish to find out the total acceleration in the modulus function, we have the following equation:

 

[vec{a_{(total)}} = |vec{a_{(total)}}| = sqrt{a_r^2}+a_t^2] 

So, the total acceleration is the square root of the sum of the squares of the radial and tangential acceleration.

 

What are the possibilities for the Value of Tangential Acceleration?

The result of tangential acceleration may have the following three possibilities:

1. The value of tangential acceleration can be greater than zero, that is, positive. This generally happens when the magnitude of the velocity vector increases with time, that is, the body has accelerated motion.

2. The value of tangential acceleration can be less than zero, that is, negative. This happens when the magnitude of the velocity vector decreases with time, that is when the object has slowed or decelerated motion.

3. The value of tangential acceleration can be equal to zero. This happens when the magnitude of the velocity vector remains constant, that is, the object is in uniform motion.

 

Summary of Tangential Acceleration

The object has a constant speed when it is executing circular motion around a circle of fixed radius in case of uniform circular motion. But if in case, the speed of the object is not constant, this is, changing with time then there will be an additional acceleration applied on the o
bject, that is known as tangential acceleration, and is applied in the direction of the tangent of the circle. The tangential acceleration is calculated as the product of the radius of the circle and angular acceleration. The rate of change of angular velocity is denoted by angular acceleration. The centripetal acceleration is the radial component of the acceleration and is calculated as the square of velocity divided by the radius of the circle. Centripetal acceleration is directed towards the center of the circular path, that is, radially inwards. Both these accelerations, that is, centripetal acceleration and tangential acceleration are mutually perpendicular to each other. Therefore, the total acceleration acting on a particular object is the sum of both the tangential acceleration and the centripetal acceleration.

 

For example, a particle moving in a circular path of radius 2m has a linear speed equal to the square of time in seconds. The linear speed is given in meters per second. Calculate the radial acceleration and the tangential acceleration on the particle at t = 2s. Also, compute the net acceleration acting on a particle.

Solution: Since, the linear speed of the particle is the square of time in seconds, therefore, the linear speed of the particle at t = 2s will be the square of 2, that is, 4 meters per second. 

The radial acceleration of a particle is calculated as the square of velocity divided by the radius of the circle. Since the velocity as calculated above is 4 m/s, therefore the square of 4 will be 16m/s and the radius of the circle given in the question is 2m. Thus, radial acceleration will be 16 divided by 2, that is, 8 meters per second square.

 

The tangential acceleration of a particle is calculated by differentiation of velocity divided by time. Since the velocity is square of time, therefore on differentiation, we will get 2t. Putting the value of t as 2sec, the acceleration will be the product of 2 and 2, which is 4 meters per second square.

 

Now, for calculating the net acceleration, under the center root of the sum of squares of both the acceleration will be taken. On calculating, this will come out to be 8.944 meters per second square.

Every object with both a magnitude and a direction is referred to as a vector.

A vector can be drawn geometrically as a guided line section with an arrow representing the direction and a length equal to the magnitude of the vector. From the tail to the head, the vector’s orientation is shown. We’ll go over the definition of a vector and some vector formulas with examples in this subject. Let’s take a look at the idea!

Vector Formula 

The Concept of Vector Formula 

In mathematics, a vector is a representation of an object that includes both magnitude and direction.

If two vectors have the same direction and magnitude, they are the same. This means that if we take a vector and transfer it to a different place, we get a new vector. The vector we get at the end of this phase looks like this, and it’s the same vector we had at the start.

In physics, vectors that represent force and velocity are two common examples of vectors. Power and velocity are both acting in the same way. The magnitude of the vector would mean the force’s intensity or the velocity’s related speed. Since displacement is directly attached to distance, distance and displacement are not the same.

An arrow mark is commonly used to represent a vector.

Also, whose length is proportional to the magnitude and whose direction is the same as the quantity. Scaled vector diagrams with values are often used to describe vector quantities. A displacement vector will be described in the vector diagram.

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Some Important Definitions and Vector All Formula

Vector Formula Mathematics

Magnitude

The magnitude of a vector is the length of the vector it is used in the vector formula. The magnitude of the vector a is denoted by |a| For a two-dimensional vector a = (a[_{1}], a[_{2}]), the formula for its magnitude is 

|a| = [sqrt{a_{1}^{2} + a_{2}^{2}}]

And for three-dimensional vector a = (a[_{1}], a[_{2}], a[_{3}]), the formula for its magnitude is 

|a| = [sqrt{a_{1}^{2} + a_{2}^{2} + a_{3}^{2}}]

Direction

A vector’s direction is often expressed as a counterclockwise angle of rotation around its “tail” from due East.

A vector with a direction of 30 degrees is a vector that has been rotated 30 degrees, counterclockwise relative, to due east using this convention.

Vector Formula Physics

Force 

The vector sum of two or more forces is represented by a resultant force, which is a single force.

Like two forces of magnitudes F1 and F2 function on a particle, the effect is as follows:

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Velocity

The rate of change of an object’s direction is represented by a velocity vector.

The magnitude of a velocity vector indicates an object’s speed, while the vector direction indicates the object’s direction.

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Triangular Law of Additions 

The triangle law of vector addition states that when two vectors are represented as two sides of a triangle of the same order of magnitude and direction, the magnitude and direction of the resulting vector is represented by the third side of the triangle.

As two forces, Vector A and Vector B, function in the same direction, the resulting R is the sum of the two vectors.

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The formula for Triangular law of addition: [bar{R}] = [bar{A}] + [bar{B}] 

 Parallelogram Law of Addition

When two powers, A Vector B Vector formula, are expressed by the parallelogram’s opposite sides, the resultant is represented by the diagonal of a parallelogram taken from the same position.

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The formula for Triangular law of addition: [bar{R}] = [bar{A}] + [bar{B}] 

Vector Subtraction

If two powers, Vector A and Vector B, are acting in the opposite direction, The variance between the two vectors is then used to describe the resultant R.

As a result, the Vector Subtraction formula is  [bar{R}] = [bar{A}] – [bar{B}] 

Note: Any of the concepts and formulae discussed in this vector formula sheet can come in useful when learning about three-dimensional geometry.

A 3D Geometry vector formula sheet is also available on every website.

Examples of Vector Formula

Q.1) Find the Addition and Subtraction of Given Vectors.

1. (2,3,4) and (5,7,8)

2. (6,3,2) and (7,5,3)

Answer:

By using the triangular law of addition the given vectors are,

a) (2,3,4) and (5,7,8)

⇒ {2+5,3+7,4+8}

⇒ {7,10,12}

b) (6,3,2) and (7,5,3)

⇒ {6+7,3+5,2+3}

⇒ {13,8,5}

By using the vector subtraction law the given vector is,

a) (2,3,4) and (5,7,8)

⇒ {2-5,3-7,4-8}

⇒ {-3,-4,-4}

b) (6,3,2) and (7,5,3)

⇒ {6-7,3-5,2-3}

⇒ {-1,-2,-1}

A parent nucleus splits into two or more daughter nuclei to reach the stage of stability. So the unstable nucleus is considered radioactive and while splitting, it loses energy in the form of radiation.

So the whole process initiating from subdivision to loss of energy is the radioactive decay.

The three most common types of decays are:

There are certain radioactive equations for three of these, say, for the gamma decay process, we have the gamma decay formula, proceeding with this, we have the activity of radioactive substance formula. 

Also, the formula for half-life decay helps us determine the time needed for half of the original population of radioactive atoms to decay, which we will understand with the help of the radioactive half-life formula.

Radioactive Decay Equation

As per the activity of radioactive substance formula, the average number of radioactive decays per unit time or the change in the number of radioactive nuclei present is given as:

                                A   =  –  dN/dt

Here,

Also, we understand the following key points from the above radioactivity equation:

 

• The total activity relies entirely on the number of nuclei present, as we can see A dN….(a)

• During the radioactive decay, A decreases with time, as we can see that A ∝ 1/dt…..(b)

Below is the graph representing (a) and (b) definitions:

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Now, let us have a look at alpha, beta, and gamma radioactive equations.

Radioactivity Formula

In the years 1899 and 1900, a British Ernest Rutherford (working at McGill University in Montreal, Canada) and the French Physicist named Paul Villard (working in Paris) did experimental investigations on electromagnetic radiation and separated them into three kinds.

Further, Rutherford named them alpha, beta, and gamma rays depending on the penetration of matter and deflection by a magnetic field.

Here, we will talk about the following three radioactive equations:

• Alpha decay formula

• Beta decay formula

• Gamma decay formula

Alpha Decay Formula

Alpha decay results in the emission of α-particles from the radioactive nucleus.

For example, the alpha decay of  92U238 into   90Th234  is as follows:

 

In the above radioactive decay formula, we notice the following things:

1. The Uranium nucleus emits an α-particle, and therefore, its mass and charge reduced, shown in the following equation:

                 Mass number:  238 –  4 = 234 

                 Charge number: 92 – 2 = 90        

Following this, a new element formed is Thorium (Th).

In general, the radioactivity equation (1) can be represented as:

zXA   ⇾   z – ₂YA-2  +  ₂He⁴ + Q  

Also, we see that after a spontaneous α-decay process, the total mass of  90Th234 and  2He 4 was less than 92U238.          

This means, the total mass-energy (Q) also decreases, equivalent to the difference between the Initial mass-energy and the final mass-energy, stated as:

                             Q = (mx – m – mHe)     

                              Q = (mx – my – mHe). c2   

In this equation, Q is the disintegration energy, which is shared by the daughter nucleus ‘y’ and an alpha particle He.

So, the alpha particle emission occurs in the following manner:

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Beta Decay Formula

Beta decay of Thorium, i.e.,  90Th234 emits a β-particle, where the mass number of the

daughter nucleus remains invariant, while the charge increments by 1, therefore, a new element Palladium 91Pa234  forms in the following way:

90Th234 ⇾  91Pa234 + -1e0 (β-particle)

[]                              

Here,

2. The mass number of Palladium is:

               234 – 0 = 234

Its charge number becomes 91 (90+1).

The general radioactivity equation for the beta decay process is:

Z XA ⇾  z+1YA + -1e0 + Q 

 Q = the energy released in β-decay.

Gamma Decay Formula

We know that gamma rays are emitted during the decay of radioactive atomic nuclei and definite subatomic particles. 

These powerful rays are produced by the hottest and most energetic objects in the universe and are present in the electromagnetic spectrum.

 Below is the gamma decay equation of Technetium-99m to Technetium-99:

 43Tc99m      →      43Tc99      +   0γ0   (Gamma radiation) ….(2)

Another e
xample that initiates from the β-decay of 27Co60 turning into an exciting 28Ni60 nucleus, the radioactive decay equation for the same with the energy released is as follows:

27Co60  ⇾ 28Ni60**   + -1e0……(3)

So, when this exciting nucleus reaches the ground state, as a result, gamma rays are emitted with a release of energy in Mega electron Volts.

  28Ni60** ⇾ 28Ni60* + Eγ ( = 1.17 MeV)……(4)

  28Ni60* ⇾ 28Ni + Eγ (= 1.33 MeV)……(5)

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Radioactive Half Life Formula

The formula for half-life decay is:

[N(t)=N_{0}(frac{1}{2})^{frac{t}{t_{1/2}}}]……(6)

Here,

• N (t) is a function of time, which shows the amount of substance remaining after the decay in a given time.

• N is the initial quantity of the substance

• t  is the time elapsed, and

• t1/2  is the half-life of the decaying component

Definition of the Half-Life:

When half of the radioactive atom undergoes the decay process, the time needed for a quantity to reduce to half of its initial value is the half-life. When talking about the decay of half of the radioactive atoms, the time taken is the radioactive half-life. 

For this, we have a radioactive half-life formula:

[t_{1/2}=frac{0.693}{lambda }]

Here, λ is the decay constant. 

Now, let us understand the decay constant formula:

Let’s suppose that ‘N’ is the size of a population of radioactive atoms at a given time ‘t,’ and dN is the amount by which the population of the radioactive atom decreases in time dt; therefore,  the rate of change is given by the following equation:                          

dN/dt     = – λ N,  (λ  = decay constant)

Radioactive Half-Life Table

Radioactive Half-Life Graph

The graph of the above table is as follows:

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Radioactive Half-Life Formula Derivation

Let’s describe equation (6) in an exponential form:

[N(t)=N_{0}(frac{1}{2})^{frac{t}{t_{1/2}}}]

[N(t)=N_{0}e^{frac{-t}{tau}}]

Here,

τ =  A mean lifetime of the decaying quantity, which is positive 

λ is also positive

The three parameters (t1/2, τ , and λ are all directly related to each other in the following way:

[t_{1/2}=frac{ln(2)}{lambda }]  =   (ln (2). τ

We know that the value of the natural logarithmic of 2 is 0.693, so rewriting equation (7):

[t_{1/2}=frac{0.693}{lambda }]

This is the required radioactivity formula for radioactive half-life.

Conclusion

A phenomenon in which a ‌heavy‌ ‌unstable element‌ ‌disintegrates‌ ‌itself‌ into two or more daughter nuclei ‌without‌ ‌being‌ forced‌ ‌by‌ ‌any‌ ‌external‌ ‌agent‌ ‌to‌ ‌do‌ ‌so.‌ The process involves the emission of the following particles:

• α particle

• β particle

• 𝛾 particle