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1. Heat loss through an insulated pipe

Homework Statement A pipe of radius R is maintained at temperature T. It is covered in insulation and the insulated pipe has radius r. Assume all surfaces lose heat through Newton's law of cooling \vec{J} = \vec{h} \Delta T, where the magnitude h is assumed to be constant. Show that...
2. Charged particle between two conducting plates

Homework Statement Two large conducting plates are separated by a distance 'L', and are connected together by a wire. A point charge 'q' is placed a distance 'x' from one of the plates. Show that the proportion of the charge induced on each plate is 'x/L' and '(L-x)/L'. (Hint: pretend the...
3. Show that A is an orthogonal matrix

Homework Statement If {aj} and {bj} are two separate sets of orthonormal basis sets, and are related by ai = \sumjnAijbj Show that A is an orthogonal matrix Homework Equations Provided above. The Attempt at a Solution Too much latex needed to show what I tried...
4. Charge placed between two plates

Homework Statement A sheet of charge +Q is in between two large plates of a capacitor. The plates are separated by a distance 'S', and are connected to each other by a wire. The sheet of charge is closer to one plate, at a distance 'a' from it. Show that the ratio to +Q of the charges...
5. Related to stokes theorem

Homework Statement \nabla \times f \vec{v} = f (\nabla \times \vec{v}) + ( \nabla f) \times \vec{v} Use with Stoke's theorem \oint _C \vec{A} . \vec{dr} = \int \int _S (\nabla \times \vec{A}) . \vec{dS} to show that \oint _c f \vec{dr} = \int \int _S \vec{dS} \times \nabla...
6. Vector fields

Hey - I'm stuck on a concept: Are ALL vector fields expressable as the product of a scalar field \varphi and a constant vector \vec{c}? i.e. Is there always a \varphi such that \vec{A} = \varphi \vec{c} ? for ANY field \vec{A}? I ask because there are some derivations from...
7. Dipole term in a quadrupole expansion

Homework Statement There are three charges arranged on the z-axis. Charge +Q_2 at the origin, -Q_1 at (0,0,a) and -Q_1 at (0,0,-a). Using spherical polar coordinates (i.e the angle \vartheta is between r and the positive z-axis), find the potential at a point with a distance r from the...
8. Energy dissipated by air resistance through integration

Homework Statement Hey guys, I'm doing a problem on quadratic drag and energy dissipation. Basically, the question asks me to find the energy dissipated when a ball of mass 'm' is thrown directly upwards with a velocity 'v0', during the upwards journey to its maximum height, 'h'. The resistive...