Consider a solenoid (n turns/unitlength, radius R) carrying a current that follows
$$I(t)=kt$$ k constant. Calculate the Poynting vector and use it to show that the flow of energy (per unit length) into the volume occupied by the solenoid is given by,

$$\frac{\mathrm{d}W}{\mathrm{d}t}=\frac{\mathrm{d}{\mathrm{d}t}(\frac{1}{2}LI^2)$$ where L is the self-inductance per unit length of the solenoid.

Question

Consider a solenoid (n turns/unitlength, radius R) carrying a current that follows
$$I(t)=kt$$ k constant. Calculate the Poynting vector and use it to show that the flow of energy (per unit length) into the volume occupied by the solenoid is given by,

$$\frac{\mathrm{d}W}{\mathrm{d}t}=\frac{\mathrm{d}{\mathrm{d}t}(\frac{1}{2}LI^2)$$ where L is the self-inductance per unit length of the solenoid.

anonymous · Answer

Sorry that should be, $$\frac{\mathrm{d}W}{\mathrm{d}t}=\frac{\mathrm{d}}{\mathrm{d}t}(\frac{1}{2}LI^2)$$

nikvist · Answer

attachment

anonymous · Answer

Hm when I go through I get $$P=(\mu_0 n \dot{I} r/2)(\mu_0 n I)=\mu_0^2...$$ This then follows through and we have extra mu's in the final statment. the mu squared seems to come from us deriving E from B which carries extra mu.


NVM I realized I wasn't using H in calculating P.