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| | | ### Statement |
| | | |
| | | $14.3.11.$ |
| | | |
| | − | How many times will the potential difference and capacitance of a long cylin-drical capacitor change when it moves at a speed βc along the axis? |
| | | |
| | − | ### Solution |
| | + | ### Solution |
| | | |
| | − | Reference frame and geometry |
| | + | Rest frame S' |
| | | |
| | − | S': system at rest with respect to the capacitor. |
| | − | S: laboratory system, in which the capacitor moves with velocity v along the z-axis (axis of the cylinder). |
| | + | In S', the sphere is at rest, and the charge is uniformly distributed: |
| | | |
| | − | The capacitor consists of two coaxial cylinders of radii a and b (b > a) and proper length L' (very large compared to the radii). In S' the electric field is purely radial and there is no magnetic field. |
| | + | $\sigma' = \frac{Q}{4\pi R^2} \quad (\text{uniform surface density})$. |
| | | |
| | − | For a boost along z, the components of the fields transform according to: |
| | + | The electric field just outside the surface is radial and is (in CGS units): |
| | | |
| | − | $\begin{aligned} |
| | − | E_\parallel' &= E_\parallel, \quad B_\parallel' = B_\parallel,\\ |
| | − | \mathbf{E}_\perp' &= \gamma (\mathbf{E}_\perp + \mathbf{v} \times \mathbf{B}),\\ |
| | − | \mathbf{B}_\perp' &= \gamma \left(\mathbf{B}_\perp - \frac{\mathbf{v}}{c^2} \times \mathbf{E}\right). |
| | − | \end{aligned}$ |
| | + | $E' = \frac{Q}{R^2}$ |
| | | |
| | − | In our case, in S' (at rest): |
| | + | There is no magnetic field in S': $\mathbf{B}' = 0$. |
| | | |
| | − | $\mathbf{E}'$ |
| | − | is radial (perpendicular to z), therefore $\mathbf{E}_\perp' = \mathbf{E}', E_\parallel' = 0$ |
| | − | $\mathbf{B}' = 0$ |
| | + | Transformation to the laboratory frame S |
| | | |
| | − | In S |
| | − | using the inverse transformation (changing v \to -v) : |
| | | |
| | − | $\mathbf{E}_\perp = \gamma (\mathbf{E}_\perp' - \mathbf{v} \times \mathbf{B}') = \gamma \mathbf{E}_\perp', \quad \mathbf{E}_\parallel = 0$ |
| | + | The laboratory frame S moves relative to S' with velocity $\mathbf{v} = -v\hat{\mathbf{z}}$ |
| | + | Equivalently, the sphere moves in S with velocity $\mathbf{v} = v\hat{\mathbf{z}} (v = \beta c)$. |
| | | |
| | − | $\mathbf{B}_\perp = \gamma \left(\mathbf{B}_\perp' + \frac{\mathbf{v}}{c^2} \times \mathbf{E}'\right) = \gamma \frac{\mathbf{v}}{c^2} \times \mathbf{E}'$ |
| | + | The components of the electric field parallel and perpendicular to $\mathbf{v}$ transform as: |
| | | |
| | − | The radial electric field in S is: |
| | + | $E_\parallel = E'_\parallel, \qquad $ |
| | + | $\mathbf{E}_\perp = \gamma (\mathbf{E}'_\perp + \mathbf{v} \times \mathbf{B}') = \gamma \mathbf{E}'_\perp$ |
| | | |
| | − | $\boxed{E_r = \gamma E_r'}$ |
| | + | with $ \gamma = 1/\sqrt{1-\beta^2}$ |
| | | |
| | − | Potential difference between the cylinders |
| | | |
| | − | The potential difference in S' is $ |
| | − | V' = \int_a^b E_r' \, dr$ |
| | + | Electric field on the surface (in S) |
| | + | |
| | + | In S', the electric field is radial: |
| | + | |
| | + | At the equator $(\theta' = \pi/2): \mathbf{E}'$ is perpendicular to $\mathbf{v} → \mathbf{E}'_\perp = \mathbf{E}'$ |
| | + | |
| | + | At the poles $(\theta' = 0): \mathbf{E}'$ |
| | + | is parallel to $\mathbf{v} → \mathbf{E}'_\parallel = \mathbf{E}'$ |
| | + | |
| | | In S: |
| | | |
| | − | $V = \int_a^b E_r \, dr = \int_a^b \gamma E_r' \, dr = \gamma V'$ |
| | + | Equator:$ E = \gamma E' = \dfrac{\gamma Q}{R^2} → maximum$ |
| | | |
| | − | The transverse distances (a and b) do not change because the motion is parallel to the axis. |
| | + | Poles:$ E = E' = \dfrac{Q}{R^2} → minimum$ |
| | | |
| | − | $\boxed{V = \gamma V' = \frac{V'}{\sqrt{1-\beta^2}}}$. |
| | + | Thus: |
| | | |
| | − | The potential difference increases by a factor of $ \gamma$ |
| | + | $\boxed{E_{\max} = \frac{Q}{R^2\sqrt{1-\beta^2}}}, \qquad$ |
| | | |
| | − | Capacitance |
| | + | $E_{\min} = \frac{Q}{R^2}$ |
| | | |
| | − | The total charge Q on the plates is a relativistic invariant (scalar). In S' the capacitance is $C' = Q/V'$ |
| | − | In S: |
| | + | Surface charge density (in S) |
| | | |
| | − | $C = \frac{Q}{V} = \frac{Q}{\gamma V'} = \frac{C'}{\gamma}$ |
| | + | The total charge Q is an invariant. In S, the sphere appears as an ellipsoid with semi-axes R, R, $R\sqrt{1-\beta^2}$ |
| | + | (contracted in the direction of motion). |
| | + | The surface density $\sigma$ is obtained from the transformation of the four-current or directly from charge conservation when transforming area elements. |
| | | |
| | − | $\boxed{C = C' \sqrt{1-\beta^2}}$ |
| | + | Equator (normal perpendicular to $\mathbf{v}): \sigma_{\max} = \gamma \sigma'$ |
| | + | · Poles (normal parallel to $\mathbf{v}): \sigma_{\min} = \sigma'/\gamma$ |
| | | |
| | − | The capacitance decreases by a factor of \gamma (it is multiplied by |
| | − | $\sqrt{1-\beta^2})$ |
| | + | Therefore |
| | + | $\sigma_{\max} = \frac{Q}{4\pi R^2\sqrt{1-\beta^2}}, \qquad$ |
| | | |
| | − | #### Answer |
| | + | $\sigma_{\min} = \frac{Q\sqrt{1-\beta^2}}{4\pi R^2}$ |
| | | |
| | − | $\boxed{C = C' \sqrt{1-\beta^2}}$ |
| | + | ### ANSWER |
| | | |
| | − | $\boxed{V = \gamma V' = \frac{V'}{\sqrt{1-\beta^2}}}$. |
| | + | $\boxed{E_{\max} = \frac{Q}{R^2\sqrt{1-\beta^2}}}, \qquad$ |
| | + | |
| | + | $\boxed{\sigma_{\max} = \frac{Q}{4\pi R^2\sqrt{1-\beta^2}}}, \qquad$ |
| | + | |
| | + | $\boxed{\sigma_{\min} = \frac{Q\sqrt{1-\beta^2}}{4\pi R^2}}$ |
