Physics Vidyapith ✍️

Mathematical Analysis of magnetic field at the center of circular loop: Let us consider, a current-carrying circular loop of radius $a$ in which $i$ current is flowing. Now take a small length of a current element $dl$ so magnetic field at the center of a circular loop due to the length of current element $dl$. According to Biot-Savart Law: $dB=\frac{\mu_{\circ}}{4 \pi} \frac{i .dl .sin\: \theta}{a^{2}} $ Here $\theta$ is the angle between length of current element $\left( \overrightarrow{dl} \right)$ and radius $\left( \overrightarrow{a} \right)$. These are perpendicular to each other i.e. $\theta = 90^{\circ}$ $dB=\frac{\mu_{\circ}}{4 \pi} \frac{i .dl .sin\: 90^{\circ}}{a^{2}} $ $dB=\frac{\mu_{\circ}}{4 \pi} \frac{i .dl }{a^{2}} \qquad \left(1 \right)$ The magnetic field at the center due to a complete circular loop $B=\int dB \qquad \left(2 \right)$ From equation $(1)$ and equation $(2)$ $B=\int \frac{\mu_{\circ}}{4 \pi} \frac{i .dl}{a^{2}}$ $B

Ampere's Circuital Law Statement: When the current flows in any infinite long straight conductor then the line integration of the magnetic field around the current-carrying conductor is always equal to the $\mu_{0}$ times of the current. $\int \overrightarrow{B}. \overrightarrow{dl} = \mu_{\circ} i$ Derivation of Ampere's Circuital Law: Let us consider, An infinite long straight conductor in which $i$ current is flowing, then the magnetic field at distance $a$ around the straight current carrying conductor $B=\frac{\mu_{\circ}}{2 \pi} \frac{i}{a}$ Now the line integral of the magnetic field $B$ in a closed loop is $\oint \overrightarrow{B}. \overrightarrow{dl} = \oint \frac{\mu_{\circ}}{2 \pi} \frac{i}{a} dl$ $\oint \overrightarrow{B}. \overrightarrow{dl} = \frac{\mu_{\circ}}{2 \pi} \frac{i}{a} \oint dl$ $\oint \overrightarrow{B}. \overrightarrow{dl} = \frac{\mu_{\circ}}{2 \pi} \frac{i}{a} \left(2 \pi a \right) \qquad \left( \because

Derivation of Force between two long and parallel current-carrying conductors: Let us consider: The two long straight, parallel conductors = $PQ$ and $RS$ The length of the conductor= $l$ The distance between the parallel conductor = $r$ The current flowing in conductor $PQ$ = $i_{1}$ The current flowing in conductor $RS$ = $i_{2}$ The magnetic field due to conductor $PQ$ = $B_{1}$ The magnetic field due to conductor $RS$ = $B_{2}$ The magnetic force on conductor $PQ$= $F_{1}$ The magnetic force on conductor $RS$= $F_{2}$ Force Between Parallel Current Carrying Conductor Now Consider the magnetic force on conductor $RS$ is i.e. $F_{2}=i_{2}B_{1}l sin\theta$ Where $\theta$ is the angle between the magnetic field and length element of conductor i.e. $\theta=90^{\circ}$ so above equation can be written as, $F_{2}=i_{2}B_{1}l sin 90^{\circ}$ $F_{2}=i_{2}B_{1}

Derivation of force on current-carrying conductor in uniform magnetic field: Let us consider: The length of the conductor - $l$ The cross-section area of the current carrying conductor - $A$ The current flow in a conductor- $i$ The drift or average velocity of the free electrons - $v_{d}$ The current-carrying conductor is placed in a magnetic field - $B$ The total number of free electrons in the current carrying conductor - $N$ Force on current carrying conductor in the uniform magnetic field Now the magnetic force on one free electron in a conductor - $F'= ev_{d}B sin\theta \qquad(1)$ The net force on the conductor is due to all the free electrons present in the conductor $F=N\: F' \qquad(2)$ Let $N$ is the number of free electrons per unit volume of conductor. So the total number of free electrons in the $Al$ volume of the conductor will be $N=nAl \qquad(3)$ Now substitute the value of $N$ and $F'$ from above equation $(1)$ and equat

Current Carrying Solenoid: The Solenoid is an artificial magnet which is used for different purposes. Principle of Solenoid: The principle of the solenoid is based on the "Ampere Circuital Law" and its magnetic field is raised due to the current carrying a circular loop. Construction of Solenoid: The current carrying solenoid is consist of insulated cylindrical material and conducting wire. The conducting wire like copper is wrapped closely around the insulated cylindrical material ( like cardboard, clay, or plastic). The end faces of the conducting wire are connected to the battery. Long Current Carrying Solenoid Working: When the electric current flow in the solenoid then a field (i.e. Magnetic field) is produced around and within the current carrying solenoid. This magnetic field is produced in solenoid due to circular loops of the solenoid and the direction of the magnetic field is depend upon the direction of the electric current flow i

Derivation→ Let us consider, A circular loop which have radius $a$, carrying a current $i$. Let take a point $P$ on the axis of the loop at a distance $x$ from the center $O$ of the loop. Now to find the magnetic field at point $P$, take a small current element of length $dl$ at the top of the loop. Let $r$ is the distance between the current element and point $P$. Now apply the Biot-Savart's Law to find the magnitude of the magnetic field due to small current-element $dl$ at point $P$ i.e. $dB=\frac{\mu_{\circ}}{4 \pi} \frac{i \: dl\: sin\theta}{r^{2 }}$ Where $\theta$ is the angle between the length of the element $dl$ and the line joining the element to the point $P$. Here this angle is $90^{\circ}$ $dB=\frac{\mu_{\circ}}{4\pi} \frac{i \: dl}{r^{2}}\qquad(1)$ The direction of the magnetic field $d\overrightarrow{B}$ is in the plane of the paper and at right angles to the line $r$ as shown in the figure below (By applying the right-hand screw rule). It can be

Derivation→ Let us consider, a current-carrying conductor $XY$ having length $l$ in which current $i$ is flowing from $X$ to $Y$. Now, To find the magnetic field due to the conductor, take a point $P$ at a distance $d$ from point $O$ of the conductor. Now consider a small length element $dl$ at the conductor which is making an angle $\theta$ from point $P$. The length element $dl$ is also making the angle $d\theta$ from point $O$. If $\theta_{1}$ and $\theta_{2}$ is the angle from point $O$ to point $X$ and $Y$ respectively. Than magnetic field at point $P$ due to small length element $dl$ which is at a distance $r$ $dB=\frac{\mu_{0}}{4\pi} \frac{i\: dl \: sin (90+\theta)}{r^{2}}$ $dB=\frac{\mu_{0}}{4\pi} \frac{i \: dl \: cos\theta}{r^{2}} \qquad(1)$ Magnetic Field due to a Straight Current-Carrying Conductor of Finite Length But from the figure, In $\Delta NOP$ $cos\theta=\frac{d}{r}$ $r=\frac{d}{cos\theta} \qquad(2)$ $tan\theta=\frac{l}{d}$

Description: The motion of any particle is depends upon the force applying on it. In the magnetic field, the motion of charge is perpendicular to the magnetic force applied on the charge particle because of that the path of the charge particle becomes circular. If the direction of force and the direction of motion are not perpendicular to each other that is they are at any angle then the path of the charge particle becomes becomes helical. A.) When $\overrightarrow{v}$ is perpendicular to $\overrightarrow{B}$→ Let us consider a charge $q$ enters in the magnetic field $\overrightarrow{B}$ from a point $O$ with velocity $\overrightarrow{v}$ directed perpendicular to the magnetic field $\overrightarrow{B}$.The general Expression for the force acting on the particle is $\overrightarrow{F}=q(\overrightarrow{v} \times \overrightarrow{B})$ Here the magnetic field $\overrightarrow{B}$ is perpendicular to the plane of the page directed downwards which is s

Biot-Savart Law: Biot-Savart law was discovered in 1820 by two physicists Jeans-Baptiste Biot and Felix Savart. According to this law: The magnetic field is directly proportional to the length of the current element. $dB \propto dl \qquad (1)$ The magnetic field is directly proportional to the current flowing in the conductor. $dB \propto i \qquad (2)$ The magnetic field is inversely proportional to the square of the distance between length of the current element $dl$ and point $P$ (This is that point where the magnetic field has to calculate). $dB \propto \frac{1}{r^{2}} \qquad (3)$ The magnetic field is directly proportional to the angle of sine. This angle is the angle between the length of the current element $dl$ and the line joining to the length of the current element $dl$ and point $P$. $dB \propto sin\theta \qquad (4)$