Moving Charges & Magnetism – Comprehensive Notes

Historical Background and How Electricity and Magnetism Were Unified

• For over 2000 years, people studied electricity and magnetism separately. They were finally linked in the 1820s.

• In 1820, Hans Christian Ørsted saw a compass needle move when placed near a wire carrying electric current.

  • The needle lined up in a circle around the wire. If he reversed the current, the needle pointed the other way.

  • Iron filings sprinkled nearby also formed perfect circles, showing the shape of the magnetic field.

  • Conclusion: moving electricity (current) creates a magnetic field.

• In 1864, James Clerk Maxwell wrote down a set of laws that united electricity and magnetism, predicting that light itself is an electromagnetic wave.

• Later in the 1800s, Hertz found radio waves, and people like J.C. Bose and Marconi learned how to make and detect them, leading to a boom in 20th-century technology.

Symbols and How We Show Direction

• When electricity or a magnetic field comes out of the page, we use a dot $<br>T$ (like the tip of an arrow).

• When it goes into the page, we use a cross $<br>V$ (like the back of an arrow).

• We'll use these symbols often when talking about magnetism.

Magnetic Force and the Lorentz Force

• Remember from electric fields: an electric field $<br>E$ pulls on a charge $q$ with a force $<br>F=q<br>E$. The field itself is $<br>E=<br>frac{Q\,<br>hat{r}}{4<br>pi\varepsilon_0 r^{2}}$, where $Q$ is the source charge.

• When both electric and magnetic fields are present (Lorentz Force): The total force $<br>F$ on a charged particle $q$ moving with velocity $<br>v$ is $<br>boxed{<br>F=q\left(<br>E+<br>v\times<br>B\right)}$.

  • The magnetic part of the force, $q\,<br>v\times<br>B$, depends on the charge ($q$), its speed and direction ($<br>v$), and the magnetic field strength and direction ($<br>B$).

  • The magnetic force is zero if the particle is not moving ($<br>v=0$) or if it moves parallel to the magnetic field ($<br>v\parallel<br>B$).

  • You find the direction of this force using the right-hand rule (or screw rule).

• Measuring B (the Tesla):

  • One Tesla (T) is defined as the magnetic field strength where a 1 Coulomb charge moving at 1 meter per second perpendicular to the field feels a 1 Newton force. This means $<br>B<br>|=1\;\text{T}$ when $q=1\,\text{C},\;v=1\,\text{m/s},\;F=1\,\text{N}$ (and they are perpendicular).

  • 1\;\text{T}=\text{N\cdot{}s}/\text{C\cdot{}m}. A smaller unit, 1 Gauss, is $10^{-4}\,\text{T}$. The Earth's magnetic field is about $3.6\times10^{-5}\,\text{T}$.

Magnetic Force on a Wire Carrying Current

• For a straight wire (rod) of length $l$ carrying current $I$ in a magnetic field $<br>B$: the force it experiences is
  F=I\,\nl\times
  B (Equation 4.4). This comes from considering the number of charge carriers ($n$), their charge ($q$), and their drift speed ($v<em>d$) within the wire's cross-section ($A$): F=nAlq\,\nvd\times
  B.

• For a wire that isn't straight or is in a non-uniform field, you sum up (integrate) the force on each small piece: $<br>displaystyle <br>F=\int I\,d\boldsymbol{l}\times<br>B$.

• Example 4.1: If a 1.5-meter wire weighing 0.2 kg has a 2 Amp current and is levitated (held up) by a magnetic field, the required magnetic field strength is $B=0.65\,\text{T}$.

• Example 4.2: If the magnetic field $<br>B$ points along the +y direction and a charged particle moves along the +x direction ($<br>v\parallel +x$), then the force on an electron (negative charge) is in the -z direction, while the force on a proton (positive charge) is in the +z direction.

How a Charged Particle Moves in a Uniform Magnetic Field

• When a charged particle enters a uniform magnetic field exactly sideways (perpendicularly), it moves in a perfect circle.

  • The circle's radius is given by $r=\dfrac{mv}{qB}$.

  • The speed at which it goes around the circle (angular frequency, also called cyclotron frequency) is $\omega=\dfrac{qB}{m}$, or in regular frequency, $\nu_c=\dfrac{qB}{2\pi m}$. What's interesting is that this frequency doesn't depend on how fast the particle is moving!

• If the particle's velocity has a part that's parallel to the magnetic field, it will move in a spiral shape called a helix. The 'pitch' ($p$) of this helix (distance advanced per turn) is $p=\dfrac{2\pi m v_{\parallel}}{qB}$.

• Example 4.3: An electron moving at $3\times10^7\,\text{m/s}$ in a magnetic field of $6\times10^{-4}\,\text{T}$ will move in a circle with a radius of $0.28\,\text{m}$, at a frequency of $17\,\text{MHz}$, and it has an energy of $2.5\,\text{keV}$.

The Biot–Savart Law

• The Biot–Savart Law helps us find the magnetic field ($d<br>B$) produced by a very small piece of wire ($d\boldsymbol{l}$) carrying current ($I$) at a point P. The formula is: d
  B=\frac{\mu_0}{4\pi}\,\frac{I\,d\boldsymbol{l}\times\nhat{\mathbf{r}}}{r^{2}}.

  • The strength (magnitude) of this tiny magnetic field is $dB=\dfrac{\mu_0}{4\pi}\dfrac{I\,|d\boldsymbol{l}|\sin\theta}{r^{2}}$.

• Here, $\mu_0$ (mu-nought) is a special constant called the permeability of free space, and its value is 4\pi\times10^{-7}\,\text{T\cdot{}m/A}.

• How it's similar and different from Coulomb's Law (for electric fields):

  • Both laws show the field strength drops off with $1/r^{2}$ (meaning it gets weaker quickly with distance) and can be added up for multiple sources (superposition).

  • Source: For electricity, the source is a simple number (scalar charge). For magnetism, the source is a small current element, which has both a size and a direction (vector $I d\boldsymbol{l}$).

  • The magnetic field is always at right angles (perpendicular) to both the current element ($d\boldsymbol{l}$) and the line connecting it to the point (\nhat{\mathbf{r}}). This angle dependence is not seen in electrostatic fields.

• The constant $\mu<em>0$, along with $\varepsilon</em>0$ (from electricity), together define the speed of light in a vacuum: $c=1/\sqrt{\mu<em>0\varepsilon</em>0}$, showing the deep connection between light and electromagnetism.

• Example 4.4: For a 1 cm piece of wire (d\boldsymbol{l}=1\,\text{cm}\,\nhat{i}) carrying 10 Amperes ($I=10\,\text{A}$) and located 0.5 meters away along the y-axis (r=0.5\,\nhat{j}\,\text{m}), the tiny magnetic field produced is $4\times10^{-8}\,\text{T}$ in the +z direction (+\nhat{k}).

Magnetic Field from Specific Current Shapes

Straight Infinite Wire

• For a very long, straight wire carrying current $I$: The magnetic field ($B$) at a distance $r$ from the wire is $B=\dfrac{\mu*0 I}{2\pi r}$. The field lines form circles around the wire, and their direction can be found using the right-hand rule.

• If you are inside a thick wire of radius $a$ where current is spread evenly, the magnetic field increases linearly from the center: $B(r)=\dfrac{\mu_0 I r}{2\pi a^{2}}$ for (r<a). Outside the wire, it's the same as a thin wire.

Circular Loop (on axis)

• For a circular loop of wire with radius $R$ carrying current $I$: The magnetic field ($B<em>x$) at a point along its central axis, at a distance $x$ from the center, is given by: Bx=\frac{\mu*0 I R^{2}}{2\left(x^{2}+R^{2}\right)^{3/2}} \;\nhat{i}.

• Right at the center of the loop ($x=0$), the field is strongest and is $B=\dfrac{\mu_0 I}{2R}$.

• Magnetic field lines around a loop form closed loops. You can find their direction by curling the fingers of your right hand in the direction of the current.

• Example 4.5: A semi-circular wire carrying 12 Amperes ($I=12\,\text{A}$) with a radius of 2 cm ($R=2\,\text{cm}$) creates a magnetic field of $1.9\times10^{-4}\,\text{T}$ pointing into the page at its center.

• Example 4.6: A coil with 100 turns, a radius of 0.1 meters, and a current of 1 Ampere produces a magnetic field of $6.28\times10^{-3}\,\text{T}$ at its center.

Solenoid

• A solenoid is a coil of wire shaped like a long cylinder. If it's very long compared to its width, and has $n$ turns of wire per unit length carrying current $I$: The magnetic field ($B$) inside is strong and uniform, given by $B=\mu_0 n I$. The field outside is very weak, almost zero.

• This field strength can be proven using Ampère's Circuital Law with a rectangular 'Amperian loop'.

• Example 4.8: A solenoid 0.5 meters long with 500 turns ($N=500$) carrying an 8 Amp current ($I=8\,\text{A}$) has $n=1000\,\text{m}^{-1}$ turns per meter, resulting in an internal magnetic field of $B=6.28\times10^{-3}\,\text{T}$.

Ampère’s Circuital Law (integral form)

• Ampère's Circuital Law is a fundamental way to describe how electric currents create magnetic fields. In its integral form, it states: $<br>displaystyle \oint<em>C B\cdot d\boldsymbol{l}=\mu</em>0 I_{\text{enc}}$. This means the sum of the magnetic field (B) along any closed loop ($\oint*C$) is proportional to the total current ($I_{\text{enc}}$)
  enclosed by that loop.

• This law is especially useful for finding magnetic fields in situations with high symmetry, like a straight wire, a solenoid, or a toroid (a donut-shaped coil).

Force Between Two Parallel Currents and the Definition of Ampere

• If you have two long, parallel wires separated by a distance $d$, and they carry currents $I<em>a$ and $I</em>b$ in the same direction: They will attract each other with a force per unit length given by: $<br>boxed{\dfrac{F}{L}=\frac{\mu<em>0 I</em>a I_b}{2\pi d}}$.

• If the currents go in opposite directions, the wires will push each other away (repel).

• The Ampere (A), the unit of electric current, is officially defined using this force. If two infinitely long, parallel wires are 1 meter apart in a vacuum and each carries 1 Ampere of current, they will feel a force of 2\times10^{-7}\,\text{N\cdot{}m}^{-1} (Newtons per meter of length) between them.

• Example 4.9: The Earth's magnetic field can exert a force of about 3\times10^{-5}\,\text{N\cdot{}m}^{-1} on a 1 Ampere, 1-meter long wire.

Magnetic Dipole and Torque on a Current Loop

Torque on a Rectangular Loop

• A current loop acts like a small magnet (a magnetic dipole). It has a magnetic moment $<br>m$ defined as $<br>m=N I <br>A$, where $N$ is the number of turns, $I$ is the current, and $<br>A$ is the loop's area (its direction comes from the right-hand rule).

• When this loop is placed in a uniform magnetic field $B$ (meaning the field is the same everywhere), it experiences a twisting force, or torque ($\boldsymbol{\tau}$), given by: boxed{\boldsymbol{\tau}=\nm\times B}.

  • The strength of this torque (magnitude) is $\tau=mB\sin\theta$, where $\theta$ is the angle between the magnetic moment and the magnetic field.

  • The loop will try to align itself so its magnetic moment $<br>m$ is parallel to the magnetic field $<br>B$. This is its most stable position.

• Example 4.10: A coil with 100 turns, a radius of 0.1 meters, and a current of 3.2 Amperes has a magnetic field of $2\times10^{-3}\,\text{T}$ at its center and a magnetic moment m=10\,\text{A\cdot{}m}^2. If this coil is placed in a 2 Tesla magnetic field and rotated 90 degrees, it would experience a torque of 20\,\text{N\cdot{}m}. (Its resulting angular speed can be found using energy principles).

Current Loop as a Magnetic Dipole (Far-Away Field)

• If you look at the magnetic field from a current loop very far away (where $x\gg R$), it behaves like a magnetic dipole. Along its axis, the field strength is: $B=\dfrac{\mu_0}{4\pi}\dfrac{2m}{x^{3}}$.

• Along the line perpendicular to its axis (equatorial line), the field strength is: $B=\dfrac{\mu_0}{4\pi}\dfrac{m}{x^{3}}$.

• This is similar to how an electric dipole behaves, where electric dipole moment $p$ becomes magnetic moment $m$, and $1/\varepsilon<em>0$ becomes $\mu</em>0$.

• Unlike electricity, where single positive or negative charges exist, we have never found isolated 'magnetic charges' (magnetic monopoles). The simplest magnetic object is always a dipole, like a current loop or a bar magnet with both a north and south pole.

Moving-Coil Galvanometer (MCG)

• A Moving-Coil Galvanometer (MCG) is a device used to detect or measure small electric currents. It typically consists of: a rectangular coil with many turns and area $A$, placed in a magnetic field ($B$) that is shaped radially (often with a soft-iron core to make the field stronger and uniform), a torsion spring with a constant $k$ that tries to restore the coil to its original position, and a mirror or needle to show the deflection.

• When current passes through the coil, it creates a torque, which causes the coil to turn until the spring's opposing torque balances it. The angle of deflection ($\phi$) at equilibrium is: $\phi=\dfrac{N I A B}{k}$.

• Current sensitivity ($S_I$) tells us how much the needle deflects for a given current. It's the deflection angle per unit current: $S_I=\dfrac{\phi}{I}=\dfrac{NAB}{k}$.

• To turn it into an Ammeter (to measure larger currents), a small resistance called a 'shunt' ($r<em>s$) is connected in parallel with the galvanometer. This makes the overall resistance of the ammeter very low, close to $r</em>s$, so it doesn't significantly change the circuit it's measuring.

• To turn it into a Voltmeter (to measure voltage), a very large resistance ($R$) is connected in series with the galvanometer. This makes the total resistance of the voltmeter very high, close to $R$, so it doesn't draw much current from the circuit. Its voltage sensitivity ($S<em>V$) is given by $S</em>V=\dfrac{S*I}{R}$.

• There's a trade-off: if you double the number of turns ($N$), you double the current sensitivity ($S<em>I$). But if you also need to double the series resistance ($R$) for a voltmeter, the voltage sensitivity ($S</em>V$) remains the same.

• Example 4.12: This example shows how attaching measuring devices can affect the circuit. If an ammeter has a high internal resistance, it significantly lowers the circuit current (e.g., to 0.048 A). A properly shunted ammeter (low resistance) gives a more accurate reading (e.g., 0.99 A compared to an ideal 1.0 A current).

Key Formulas & Units

• Lorentz Force: $<br>F=q(<br>E+<br>v\times<br>B)$.

• Force on a wire: I \nl\times
  B.

• Cyclotron frequency: $\nu_c=\dfrac{qB}{2\pi m}$.

• Biot–Savart Law (for a small piece of wire): d
  B=\dfrac{\mu_0}{4\pi}\dfrac{I d\boldsymbol{l}\times\nhat{r}}{r^{2}}.

• Magnetic field from a straight wire: $B=\dfrac{\mu_0 I}{2\pi r}$; inside a uniform wire $B\propto r$.

• Magnetic field at the center of a loop: $B=\dfrac{\mu<em>0 I}{2R}$; at a point on the axis: $B</em>x=\dfrac{\mu_0 I R^{2}}{2(x^{2}+R^{2})^{3/2}}$.

• Magnetic field inside a solenoid: $B=\mu_0 n I$.

• Force per unit length between parallel wires: $F/L=\dfrac{\mu<em>0 I</em>a I_b}{2\pi d}$.

• Magnetic moment: $m=NI A$.

• Torque on a magnetic dipole: \boldsymbol{\tau}=\nm\times
  B.

• Permeability of free space: \mu_0=4\pi\times10^{-7}\,\text{T\cdot{}m/A}.

Important Ideas & Connections

• Important Idea: Magnetic field lines always form closed loops; they never start from a 'north pole' and end at a 'south pole' in the same way electric field lines start from positive charges and end on negative ones.

• Newton's third law (action-reaction) generally applies to forces between steady currents. However, for electric and magnetic situations that change over time, you need to consider the momentum carried by the fields themselves.

• The Lorentz force for magnetism depends on the particle's speed and direction. In a more advanced view (relativistic physics), electricity and magnetism are actually two different ways of looking at the same thing (a single 'electromagnetic field'), and how they appear depends on your motion; there's no single 'preferred' reference point.

• Ampère's law can be derived from the Biot-Savart law, just like Gauss's Law in electrostatics can be derived from Coulomb's Law. They are two ways to describe the same underlying physics.