Electrodynamics
- What does the Gauss divergence theorem state and how is it used in electrodynamics? In one line The divergence theorem states that the total outward flux of a vector field through a closed surface equals the volume integral... 1
- What does Stokes' theorem state and how is it used to derive Maxwell's curl equations? In one line Stokes' theorem states that the line integral of a vector field around a closed loop equals the surface integral of its curl ove... 1
- What are Maxwell's equations in point (differential) form and how are they derived? In one line Maxwell's four equations in point form are nabla·D = ρ , nabla·B = 0 , nabla×E = -∂B/∂ t and nabla×H = J + ∂D/∂ t , describing e... 1
- What are Maxwell's equations in integral form and what does each one mean physically? In one line In integral form, Maxwell's equations relate the flux of D and B through closed surfaces, and the circulation of E and H around... 1
- Why are Maxwell's equations significant and what is the role of displacement current? In one line Maxwell's equations unify electricity, magnetism and light into one theory, and Maxwell's displacement current term made them co... 1
- What is the gradient of a scalar field and what is its physical meaning? In one line The gradient of a scalar field is a vector that points in the direction in which the field increases fastest, and its magnitude... 1
- What is the divergence of a vector field and what does it tell us physically? In one line Divergence of a vector field measures the net outward flow (flux) per unit volume from a point, showing whether the point acts a... 1
- What is the curl of a vector field and what is its physical significance? In one line Curl of a vector field is a vector that measures how much the field circulates or rotates around a point, with direction along t... 1
- What does Gauss's law of electrostatics state and how is its point form obtained? In one line Gauss's law states that the total electric flux through any closed surface equals the net charge enclosed divided by ε₀ ; in poi... 1
- What is Ampere's circuital law and how is it written in differential (point) form? In one line Ampere's circuital law states that the line integral of magnetic field around any closed loop equals μ₀ times the net current pa... 1
- What does Faraday's law of electromagnetic induction state and what is its point form? In one line Faraday's law states that the emf induced in a closed loop equals the negative rate of change of magnetic flux through it; in po... 1
- State Ampere’s Circuital law. Derive Maxwell’s fourth equation in differential form. Ampere's circuital law: The line integral of the magnetic field intensity vec H around any closed path equals the net current enclosed by th... 1
- Show that the divergence of the curl of a vector is zero. Let vec A=A_xhat i+A_yhat j+A_zhat k . nabla×vec A=Big(∂ A_z/∂ y-∂ A_y/∂ zBig)hat i+Big(∂ A_x/∂ z-∂ A_z/∂ xBig)hat j+Big(∂ A_y/∂ x-∂ A_x/∂ y... 1
- Determine the curl of these vector fields. (1) \(\vec A=\hat a_x(2x^2+y^2)+\hat a_y(xy-y^2)\) (2) \(\vec A=yz\,\hat a_x+4xy\,\hat a_y+y\,\hat a_z\) nabla×vec A=beginvmatrixhat a_x&hat a_y&hat a_z\\ ∂_x&∂_y&∂_z\\ A_x&A_y&A_zendvmatrix (1) A_x=2x²+y² , A_y=xy-y² , A_z=0 x-component: ∂_yA_z... 1
- What do you mean Curl of vector. Write its significance. Curl: The curl of a vector field vec A is a vector whose magnitude is the maximum circulation of vec A per unit area, as the area shrinks to... 1
- Find Curl of vector F (x2y + xy2z + 3x2z2) at point (1,-1,1) Interpretation: The printed components are taken as vec F=x²y hat i+xy²z hat j+3x²z² hat k . nabla×vec F=beginvmatrixhat i&hat j&hat k\\ ∂_x... 1
- Deduce Maxwell’s second equation in differential and integral form. Maxwell's second equation is Gauss's law for magnetism. Integral form: Magnetic field lines are always closed loops, because isolated magnet... 1
- Show that the divergence of a curl of a vector is zero. Let vec A=A_xhat i+A_yhat j+A_zhat k . nabla×vec A=Big(∂ A_z/∂ y-∂ A_y/∂ zBig)hat i+Big(∂ A_x/∂ z-∂ A_z/∂ xBig)hat j+Big(∂ A_y/∂ x-∂ A_x/∂ y... 1
- What is gradient? If \(\phi=3(x^2y-y^2x)\), find \(\nabla\phi\) at the point (1, -2, -1). Gradient: The gradient of a scalar field φ(x,y,z) is the vector nablaφ=∂φ/∂ xhat i+∂φ/∂ yhat j+∂φ/∂ zhat k Its magnitude is the maximum rate... 1
- State and explain Maxwell’s fourth equation. Statement (Ampere–Maxwell law): The line integral of the magnetic field vec H around a closed path equals the total current, conduction plus... 1