Physics_1

Electric Charges

• Definition: Electric charge is a physical property of particles that causes them to experience a force when placed in an electric field. It can be positive or negative.

• Types of Charges:

  • Positive Charge: Carried by protons; attracts negative charges and repels other positive charges.

  • Negative Charge: Carried by electrons; attracts positive charges and repels other negative charges.

• Unit of Charge: The unit of electric charge is the coulomb (C).

• Charge Conservation: Electric charge is conserved in an isolated system; total charge before and after any process remains constant.

• Interactions:

  • Coulomb's Law: Describes the force between two charged objects. It states that the force (F) is directly proportional to the product of the charges (q1 and q2) and inversely proportional to the square of the distance (r) between them:

    F = k * (|q1 * q2| / r^2)

    where k is Coulomb's constant.

  • Like Charges vs. Unlike Charges: Like charges repel each other while unlike charges attract.

• Conductors and Insulators:

  • Conductors: Materials that allow electrical charge to flow freely (e.g., metals such as copper and aluminum).

  • Insulators: Materials that do not permit the flow of charge (e.g., rubber, glass).

• Applications of Electric Charge:

  • In circuits to power devices.

  • In electrostatic applications like photocopiers and laser printers.

  • In various technologies such as batteries and capacitors.

• Charge Quantization: Electric charge exists in discrete amounts; the smallest basic unit of charge is the charge of an electron (approx. -1.6 x 10^-19 C).

Forces Between Multiple Charges

• Introduction: When multiple charges are present in a system, the total force acting on any charge is determined by the vector sum of the forces exerted by all other charges.

• Superposition Principle: The force on a given charge due to several other charges can be calculated by adding the individual forces from each charge vectorially. If ( F_{1}, F_{2}, \ldots, F_{n} ) are the forces on charge ( q ) due to charges ( q_{1}, q_{2}, \ldots, q_{n} ), then:

  [ F_{total} = F_{1} + F_{2} + ... + F_{n} ]

• Calculating Forces:

  • Coulomb's Law must be applied for each pair of charges. For two charges ( q_{i} ) and ( q_{j} ) separated by a distance ( r ), the force due to charge ( q_{j} ) on charge ( q_{i} ) is given by:

  [ F_{i,j} = k \cdot \frac{|q_{i} \cdot q_{j}|}{r_{ij}^2} \hat{r} ]

  where:

  • ( k ) is Coulomb's constant (approximately ( 8.99 \times 10^9 ; N m^2/C^2 ))

  • ( , \hat{r} ) is the unit vector pointing from ( q_{j} ) to ( q_{i} )

• Example: If we have three charges ( q_{1}, q_{2}, q_{3} ) at the corners of a triangle, the total force acting on any charge can be computed by finding the individual forces acting on it from the other two charges and adding them vectorially.

• Types of Charge Configurations:

  • Linear Configuration: Charges are aligned in a straight line; interactions can be simpler due to symmetry.

  • Triangular Configuration: Charges at the corners of a triangle; requires vector addition for calculations.

  • Geometric Arrangements: More complex arrangements including squares, rectangles, or random placements require careful vector analysis.

• Net Force Direction: The net force on a charge can be attractive or repulsive depending on the signs of the interacting charges (like charges repel, unlike charges attract).

Continuous Charge Distribution

• Definition: Continuous charge distribution refers to the distribution of electric charge over a continuous volume, area, or line rather than being concentrated at discrete points.

• Types of Charge Distributions:

  • Volume Charge Distribution: Charge is spread throughout a three-dimensional region. The charge density is denoted by (\rho) (coulombs per cubic meter, C/m³).

  • Surface Charge Distribution: Charge is distributed over a two-dimensional surface. The surface charge density is denoted by (\sigma) (coulombs per square meter, C/m²).

  • Line Charge Distribution: Charge is distributed along a one-dimensional line. The linear charge density is denoted by (\lambda) (coulombs per meter, C/m).

• Calculating Electric Field:

  • The electric field (\mathbf{E}) due to a continuous charge distribution can be found using integration.

  • For volume charge distributions, the electric field is calculated using:

  [ \mathbf{E} = \frac{1}{4\pi\epsilon_0} \int \frac{\rho(\mathbf{r})}{r^2} \hat{r} , dV ]

  • For surface charge distributions:

  [ \mathbf{E} = \frac{1}{4\pi\epsilon_0} \int \frac{\sigma(\mathbf{r})}{r^2} \hat{r} , dA ]

  • For line charge distributions:

  [ \mathbf{E} = \frac{1}{4\pi\epsilon_0} \int \frac{\lambda(\mathbf{r})}{r^2} \hat{r} , dl ]

• Applications:

  • Continuous charge distributions are used in various applications, such as in electric field calculations for charged plates, charged wires, and charged spheres.

  • Understanding the behavior of electric fields in conductive materials and insulators often involves analyzing continuous charge distributions.

Electric Field

Definition: The electric field ( \mathbf{E}) is a vector quantity that represents the force per unit charge experienced by a small positive test charge placed in the field.

Electric Field Due to a Point Charge

• The electric field created by a point charge ( q) at a distance ( r) from the charge is given by:

  [ \mathbf{E} = \frac{1}{4\pi \epsilon_0} \frac{q}{r^2} \hat{r} ]

  where ( \epsilon_0 ) is the permittivity of free space, and ( \hat{r} ) is the unit vector pointing away from the charge.

Electric Field Lines

• Electric field lines are a visual representation of the electric field.

  • They originate from positive charges and terminate on negative charges.

  • The density of lines indicates the strength of the electric field; closer lines represent stronger fields.

  • Field lines do not cross each other.

Electric Dipole

• An electric dipole consists of two equal and opposite charges ( +q\ and -q) separated by a distance ( a).

• The dipole moment ( \mathbf{p}) is given by:

  [ \mathbf{p} = q \cdot \mathbf{d} ]

  where (\mathbf{d}) is a vector pointing from negative to positive charge.

Electric Field Due to a Dipole

• The electric field ( \mathbf{E}) at a point in the axial line of a dipole (along the dipole moment) is given by:

  [ \mathbf{E}_{axial} = \frac{1}{4\pi \epsilon_0} \frac{2p}{r^3} ]

• The electric field at a point in the equatorial line (perpendicular to the dipole moment) is:

  [ \mathbf{E}_{equatorial} = \frac{1}{4\pi \epsilon_0} \frac{p}{r^3} ]

  where (r) is the distance from the center of the dipole.

Torque on a Dipole in a Uniform Electric Field

• The torque ( \tau) experienced by a dipole in a uniform electric field ( \mathbf{E}) is given by:

  [ \tau = \mathbf{p} \times \mathbf{E} ]

  This torque tends to align the dipole moment with the electric field.

Electric Flux

Definition: Electric flux ( \Phi_E) is a measure of the electric field ( \mathbf{E}) passing through a given surface area (A). It is defined as:

[ \Phi_E = \int \mathbf{E} \cdot d\mathbf{A} ]
where d\mathbf{A} is a vector representing an infinitesimal area on the surface, directed outward.

Statement of Gauss's Theorem

Gauss's theorem states that the total electric flux ( \Phi_E) through a closed surface is equal to the charge enclosed (Q_enclosed) divided by the permittivity of free space ( \epsilon_0):

[ \Phi_E = \frac{Q_{enclosed}}{\epsilon_0} ]

Applications of Gauss's Theorem

1. Field Due to an Infinitely Long Straight Wire:

   • For an infinitely long straight wire with uniform linear charge density ( \lambda), the electric field (E) at a distance (r) from the wire is given by: [ E = \frac{\lambda}{2\pi\epsilon_0 r} ]

   • The electric field points radially outward from the wire (for positive charge).

2. Field Due to a Uniformly Charged Infinite Plane Sheet:

   • For an infinite plane sheet with uniform surface charge density ( \sigma), the electric field (E) is uniform and given by: [ E = \frac{\sigma}{2\epsilon_0} ]

   • The electric field points away from the sheet for positive charge and is the same on both sides of the plane.

3. Field Due to a Uniformly Charged Thin Spherical Shell:

   • For a uniformly charged thin spherical shell:

     • Field Inside the Shell (r < R): The electric field inside a uniformly charged thin spherical shell is zero: [ E = 0 ]

     • Field Outside the Shell (r > R): The electric field behaves as if all the charge were concentrated at the center, given by: [ E = \frac{Q}{4\pi\epsilon_0 r^2} ]where Q is the total charge of the shell and r is the distance from the center of the shell.