The Central Question

THE Central Question what is the net effect of Qi on test charge a Now net force (\vec{F}{nll}=\sum{i}\vec{F}_{\infty i}) principle of Super position (discrete charges) 03 24 05 interaction blwtwo charges depend on * separation induvidual charges Relative motion * acceleration of charge →instantaneous force? Travels coith some finite velocity? what if the velocity of all charges is close to speed of light? →Coulombs Force Law [\vec{F}{12}=\frac{1}{4\pi\epsilon{0}}(\frac{Q_{11}Q_{12}}{\gamma^{2}})\hat{\gamma}_{12}] \[\gamma_{12}\] \[\vec{F_{21}}\] ४ \[\vec{F}_{12}\] (force on 2 due to 1) \[q_{v_{j}}\] \[Q_{p}:Q_{e}=116\times1\overline{O}^{19}C\] and \[\frac{1}{4\pi\epsilon_{\circ}}=9\times10^{Q}\frac{Nm^{2}}{C^{2}}\] \[\vec{F_{32}}\] \[\leftarrow\dot{\tau}_{L}\] Now \[\vec{F}_{2}\] \[Q_{2}\] \[\vec{F}_{12}\] \[a_{11}\] →Concept of electric field \[\vec{k}=\frac{1}{4\pi\epsilon_{0}}(\frac{Q}{\gamma^{2}})\hat{\gamma}\] and \(\vec{F_{2}}=\vec{F_{12}}+\vec{F_{3}}\) (super position is due to linearity of eq) ① at origin (imp) \[q_{3}\] field lines: pictorial representation of any field → we have both vectorfields ((\overline{E};\overline{B})) or a scalar field {VINTA 02 Recorded lecture week-1 (Mathematical preliminaries) → Electric and magnetic fields are vector fields (thus we need vector algebra and vector calculus) → Derivative f(x) • now derivative (\frac{dF(x)}{dx}) how fast P(x) changes as x changes Now Consider Some sort of Scalar field. \(T(x,y,z)\) temperature field (varies across space) scalar value T: \(R^{3}\) hence scalar freld. → So now how fast this system changes / vary wat change in position (No unique ans due to multidirectional, motion possibility) So \(dT=(\frac{\partial T}{\partial x})dx+(\frac{\partial T}{\partial y})dy+(\frac{\partial T}{\partial z})dz\) Now we can write this [11(\hat{x}\frac{\partial T}{\partial x}+\hat{y}\frac{\partial T}{\partial y}+\hat{z}\frac{\partial T}{\partial z})\cdot(dx\hat{x}+dy\hat{y}] \[dT=(\vec{\nabla}T)\cdot\vec{d}\] Now \(\vec{\nabla}^{\top}\) is known as gradient of a scalar field (which itself is a vector field) \[d\Gamma=(\vec{\nabla}\Gamma)\vec{dl}\] \(d~T=|\vec{\nabla}T||\vec{dl}|cos~\theta\) maximum when \(\mathfrak{9=0}\) 66 \[cos~0=1\] move in direction of gradient to have maximum increase in field " Now \(\vec{\nabla}=\hat{x}\frac{\partial}{\partial x}+\hat{y}\frac{\partial}{\partial y}+\hat{z}\frac{\partial}{\partial z}\) • del operator (vector operator) So what kind of things can you do with a vector * mulliply with scalar + dot padt with * cross podt with another vector anothes vector Gradient (77) gives a vector divergence (V•F). gives a scalar curl (VXF) gives a vector DIVERGENCE OF A VECTOR FIELD we have a divergence in a Scalar Quantity 2 = divergence me asure 2Vx эх aVy ay + 2Vz az the spread of vector froma particulces point → A point of positive divergence is called sousce → A point of negatie divergence is called sinR. CURL OF A VECTOR FIELD solve this determinant → Curl is ameasure of rotation of field around a point. Gurl itself in a vector field. Cimp #1.) Rules V (fig) = If + Vg (Ã +B) = FA+VB 7xCÁ+B) = VIA + TB const sule Product Rules diso • (fg) = gIF + F7g simple one * * V (FT) - F(VT) + T.VF) * V. (AxB) = B. (EXA) - A (TAB) * TX (FA) = ST×A) - AX(FF) * T* (AXE) = BY)Ä - (AP)B +Á(VB) product aules all follow the concept forderivin, put 1 - axi-aj ail where each an is a xalar field (similary for Band @ ) Sample proof : V. (FÃ) = ? Now f.A = faxi + farj+ fazk (VF)•Ã FI TÁ) thus wehave # (F) = fITÃ) + A Tf) Complicaled proof 7(AB) Now AB = dabx + ayby tazbz = $