# Flash Cards

* Average velocity:* is defined as the displacement vector divided by the total time elapsed from start to finish or in math language is defined by formula:

\[v_{av-x}=\frac{\Delta x}{\Delta t}=\frac{x_f-x_i}{t_f-t_i}\]

*is the limit of the average velocity as $\Delta t$ approaches zero. In one dimension, say $x$, is defined by formula*

**Instantaneous velocity:**\[v_x=\lim_{\Delta t\to 0} \frac{\Delta x}{\Delta t}=\frac{dx}{dt}\]

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At distance $r$ from a charged particle $q$ the magnitude of electric field is given by the following formula

\[E=\frac{1}{4\pi {\epsilon }_0}\frac{\left|q\right|}{r^2}\]

**Note 1**: the strength of electric field around the charge $q$ is directly proportional to the magnitude of charge $q$ and inversely proportional with the square of distance $r$ from it.

**Note 2**:

According to Coulomb's law, the magnitude of attractive or repulsive electric force between two charged particles $q_1$ and $q_2$ is proportional to the product of magnitude of charges and inversely proportional to the distance squared $r^2$ from each other and is founded by \[F=k\frac{\left|q_1q_2\right|}{r^2}\]

Or in vector form as

\[\vec{F}=k\frac{\left|q_1q_2\right|}{r^2}\hat{r}\]

Where $\hat{r}$ is the unit vector along the line joining the particles to one another.

In SI units, $F\to \mathrm{N}$ , $r\to \mathrm{m}$, $q\to \mathrm{C}$ and $k=9\times {10}^9\ \mathrm{N.}{\mathrm{m}}^{\mathrm{2}}\mathrm{/}{\mathrm{C}}^{\mathrm{2}}$

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**Version 1 (right hand rule):** point fingers of your right hand in the direction of $\vec v$ and curl them (through the smaller angle) toward $\vec B$. Your upright thumb shows the cross product $\vec v \times \vec B$ or the magnetic force $\vec F_B$. This force is perpendicular to the plane of $\vec v-\vec B$

**Version 2 (right hand rule):** point your fingers in the direction of $\vec B$ so the thumb points toward the velocity $\vec v$, the magnetic force on a positive charge is in the outward direction of your palm.

**Note:** the magnetic force on a negative charge is in opposite direction to that given right hand rule.

To draw the magnetic field lines around a magnet or other magnetic objects, one can use the alignment of compass needle near those.

The direction of these lines, in a magnet outside it, is away from N pole toward the S pole.

Using small iron filing one can display magnetic field patterns around magnetic objects.

The compass needles using its alignment with the magnetic field find shows the direction of the magnetic fields.

When the velocity of a charged particle $\vec v$ is perpendicular to a uniform $\vec B$, the particle moves around a circle in a plane perpendicular to $\vec B$.

There is always a centripetal force in a circular path, which in this case provided by magnetic force, therefore the radius of the circular path is

\[\underbrace{qvB}_{F_B}=\frac{mv^{2}}{r} \quad \Rightarrow \quad r=\frac{mv}{qB}\]

The time required to particle travel one circle or the peri

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If a particle of charge q and velocity $\vec v$ enters a region of space occupied by magnetic field $\vec B$, which is establishes by some source, it experiences a magnetic force $\vec F_B$ given by

\[\vec F_B=q\vec v \times \vec B\]

Using definition of the cross product, we obtain its magnitude as

\[|\vec F_B|=|q|vB\, \sin \theta\]

Where $\theta$ is the smaller angle between $\vec v$ and $\vec B$.

- The space around a magnet where the forces of attraction or repulsion on a magnetic object can be detected is called magnetic field.
- Magnetic field around a magnet or other magnetic object is visualized by magnetic field lines.
- Magnetic field around a magnet or other magnetic object can be displayed by iron filling patterns.
- Magnetic field around a magnet or other magnetic object can be detected by a little compass needle at that point.
- Small compass needle is aligned parallel to the magnetic field, with the north pole of the compass shows the direction of the magnetic field at that point.
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(1) The magnitude of a magnetic force depends only on the magnitude of the charge i.e. $F\propto |q|$

(2) Magnetic force is always perpendicular to the plane containing $\vec v$ and $\vec B$.

(3) A charge moving parallel $\theta=0$ to a magnetic field experiences zero magnetic force.

(4) A charge moving perpendicular $\theta =90{}^\circ$ to a magnetic field experiences a maximum magnetic force $F_B=qvB$.

(5) The magnetic force on a positively charge particle is in opposite direction to that of a negatively charge particle i.e. $\vec F_{B(q)}=\vec F_{B(-q)}$

(1) Electric force does work on a rest or moving charge but magnetic force does only work on moving charge.

(2) The electric force vector is along the direction of electric field $\vec E$, whereas the magnetic force is perpendicular to the plane of $\vec v-\vec B$.

(3) The electric force can change the speed or kinetic energy of a particle but magnetic force can alter only the direction of the velocity of particle.

The net electric field or force of a group of point charges at each point in space is the vector sum of the electric fields due to the individual charges at that point. in the mathematical form is written as

\[{\vec{E}}_{net}={\vec{E}}_1+{\vec{E}}_2+\dots +{\vec{E}}_n\]

To find the electric field at each point in vicinity of a charged particle $q$, place a small and insignificant positive charge, called test charge, $q_0$ at that point and then measure the force $\vec{F}$acting on it. The electric field $\overrightarrow{E}$ due to that charged point charge $q$ is defined as

\[\vec{E}=\frac{\vec{F}}{q_0}\]

Electric field is a vector quantity that its magnitude is $E=F/q_0$ and its direction is in the same direction as the force acting on the test charge. In the other words, electric field points in opposite direction of the electric force acting on a negative charged particle.