We develop a new approach to the calculation of the synchrotron radiation in a toroidal vacuum chamber. Using a small parameter a=R p , where a is the characteristic size of the cross section of the toroid and R is the bending radius, we simplify Maxwell's equations assuming that the characteristic frequency of the modes ! c=a and neglect terms of higher order in . For a rectangular cross section of the waveguide, we find an analytical solution of the equations and analyze their asymptotics at very high frequency. We then obtain an equation which gives radiation into each synchronous mode. We demonstrate the flexibility of the new method by calculating the frequencies and the loss factors for the lowest modes in square and round waveguides.

The prospect of stabilization of the $m=1$ ``rigid'' ballooning mode in an open axially symmetric long-thin trap with the help of a conducting lateral wall surrounding a column of isotropic plasma is studied. It was found that for effective wall stabilization, the beta parameter $\beta$ must exceed some critical value $\beta_{\text{crit}}$. The dependence of $\beta_{\text{crit}}$ on the mirror ratio, radial pressure profile, axial profile of the vacuum magnetic field, and the width of vacuum gap between plasma and lateral wall was studied. Minimal critical beta at the level of $70\%$ is achieved at zero vacuum gap, although stability zone at $\beta \to 1$ exists even at extremely wide vacuum gap. It is shown that when a conducting lateral wall is combined with conducting end plates simulating attachment of the end MHD stabilizers to the central cell of an open trap, there are two critical beta values and two stability zones that can merge, making stable the entire range of allowable beta values $0<\beta<1$.

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It is shown that a steepening of the radial plasma pressure profile leads to a decrease in the critical value of beta, above which, small-scale balloon-type perturbations in a mirror trap become unstable. This may mean that small-scale ballooning instability leads to a smoothing of the radial plasma profile. The critical beta values for the real magnetic field of the gas-dynamic trap and various plasma pressure radial profiles was also calculated. For a plasma with a parabolic profile critical beta is evaluated at the level of 0.72. A previous theoretical prediction for this trap was almost two times lower than maximal beta 0.6 achieved experimentally.

We study an impedance due to coherent synchrotron radiation (CSR) generated by a short bunch of charged particles passing through a dipole magnet of finite length in a vacuum chamber of a given cross section. In our method we decompose the electromagnetic field of the beam over the eigenmodes of the toroidal chamber and derive a system of equations for the expansion coefficients in the series. The general method is further specialized for a toroidal vacuum chamber of a rectangular cross section where the eigenmodes can be computed analytically. We also develop a computer code that calculates the CSR impedance for a toroid of rectangular cross section. Numerical results obtained with the code are presented in the paper.